id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-19600 | Set whose every Finite Subset is Nest is also Nest | {{explain|We have a set of sets but refer to Definition:Nest (Class Theory)}}
Let $x$ be a set of sets with the property that:
:every finite subset of $x$ is a nest.
Then $x$ is a nest. | Let $x$ be a set of sets with the given property.
Let $a, b \in x$ be arbitrary.
Then:
:$\set {a, b} \subseteq x$
and so $\set {a, b}$ is a nest.
That is:
:$a \subseteq b$ or $b \subseteq a$
As $a$ and $b$ are arbitrary, it follows that:
:$\forall a, b \in x: a \subseteq b$ or $b \subseteq a$
{{explain|As per definitio... | {{explain|We have a set of sets but refer to [[Definition:Nest (Class Theory)]]}}
Let $x$ be a [[Definition:Set of Sets|set of sets]] with the [[Definition:Property|property]] that:
:every [[Definition:Finite Subset|finite subset]] of $x$ is a [[Definition:Nest (Class Theory)|nest]].
Then $x$ is a [[Definition:Nest (C... | Let $x$ be a [[Definition:Set of Sets|set of sets]] with the given [[Definition:Property|property]].
Let $a, b \in x$ be arbitrary.
Then:
:$\set {a, b} \subseteq x$
and so $\set {a, b}$ is a [[Definition:Nest (Class Theory)|nest]].
That is:
:$a \subseteq b$ or $b \subseteq a$
As $a$ and $b$ are arbitrary, it follo... | Set whose every Finite Subset is Nest is also Nest | https://proofwiki.org/wiki/Set_whose_every_Finite_Subset_is_Nest_is_also_Nest | https://proofwiki.org/wiki/Set_whose_every_Finite_Subset_is_Nest_is_also_Nest | [
"Nests"
] | [
"Definition:Nest/Class Theory",
"Definition:Set of Sets",
"Definition:Property",
"Definition:Finite Subset",
"Definition:Nest/Class Theory",
"Definition:Nest/Class Theory"
] | [
"Definition:Set of Sets",
"Definition:Property",
"Definition:Nest/Class Theory",
"Subset Relation is Ordering",
"Subset Relation is Ordering"
] |
proofwiki-19601 | Consistency of Logical Formulas has Finite Character | Let $P$ be the property of collections of logical formulas defined as:
:$\forall \FF: \map P \FF$ denotes that $\FF$ is consistent.
Then $P$ is of finite character.
That is:
:$\FF$ is a consistent set of formulas {{iff}} every finite subset of $\FF$ is also a consistent set of formulas. | {{ProofWanted|Use Compactness Theorem. And again, the "iff" in the definition of finite character is the interesting part}} | Let $P$ be the [[Definition:Property|property]] of [[Definition:Collection|collections]] of [[Definition:Logical Formula|logical formulas]] defined as:
:$\forall \FF: \map P \FF$ denotes that $\FF$ is [[Definition:Consistent Set of Formulas|consistent]].
Then $P$ is of [[Definition:Finite Character (Property of Sets)... | {{ProofWanted|Use [[Compactness Theorem]]. And again, the "iff" in the definition of finite character is the interesting part}} | Consistency of Logical Formulas has Finite Character/Proof 1 | https://proofwiki.org/wiki/Consistency_of_Logical_Formulas_has_Finite_Character | https://proofwiki.org/wiki/Consistency_of_Logical_Formulas_has_Finite_Character/Proof_1 | [
"Logical Consistency",
"Finite Character",
"Consistency of Logical Formulas has Finite Character"
] | [
"Definition:Property",
"Definition:Collection",
"Definition:Logical Formula",
"Definition:Consistent (Logic)/Set of Formulas",
"Definition:Finite Character/Property of Sets",
"Definition:Consistent (Logic)/Set of Formulas",
"Definition:Finite Subset",
"Definition:Consistent (Logic)/Set of Formulas"
] | [
"Compactness Theorem"
] |
proofwiki-19602 | Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset | Let $\FF$ be a collection of logical formulas.
Then:
:$\FF$ be inconsistent
{{iff}}:
there exists a finite subset of $\FF$ which it itself inconsistent. | === Sufficient Condition ===
Let $\FF$ be inconsistent.
Then it is possible to assemble a proof in a finite set of statements of a contradiction.
This finite set of statements uses within it a finite subset $\GG \subseteq \FF$ of the logical formulas of $\FF$.
Hence $\GG$ is that inconsistent finite subset of $\FF$ who... | Let $\FF$ be a [[Definition:Collection|collection]] of [[Definition:Logical Formula|logical formulas]].
Then:
:$\FF$ be [[Definition:Inconsistent Set of Formulas|inconsistent]]
{{iff}}:
there exists a [[Definition:Finite Subset|finite subset]] of $\FF$ which it itself [[Definition:Inconsistent Set of Formulas|inconsis... | === Sufficient Condition ===
Let $\FF$ be [[Definition:Inconsistent Set of Formulas|inconsistent]].
Then it is possible to assemble a [[Definition:Proof|proof]] in a [[Definition:Finite Set|finite set]] of [[Definition:Statement|statements]] of a [[Definition:Contradiction|contradiction]].
This [[Definition:Finite S... | Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset | https://proofwiki.org/wiki/Set_of_Logical_Formulas_is_Inconsistent_iff_it_has_Finite_Inconsistent_Subset | https://proofwiki.org/wiki/Set_of_Logical_Formulas_is_Inconsistent_iff_it_has_Finite_Inconsistent_Subset | [
"Logical Consistency"
] | [
"Definition:Collection",
"Definition:Logical Formula",
"Definition:Inconsistent (Logic)",
"Definition:Finite Subset",
"Definition:Inconsistent (Logic)"
] | [
"Definition:Inconsistent (Logic)",
"Definition:Proof",
"Definition:Finite Set",
"Definition:Statement",
"Definition:Contradiction",
"Definition:Finite Set",
"Definition:Statement",
"Definition:Finite Subset",
"Definition:Logical Formula",
"Definition:Inconsistent (Logic)",
"Definition:Finite Sub... |
proofwiki-19603 | Consistent Set of Formulas can be Extended to Maximal Consistent Set | Let $\FF$ be a collection of consistent logical formulas.
Then $\FF$ can be extended to (that is, is a subset of) a maximal consistent set of formulas. | {{ProofWanted|This is probably the same as Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension}} | Let $\FF$ be a [[Definition:Collection|collection]] of [[Definition:Consistent Set of Formulas|consistent]] [[Definition:Logical Formula|logical formulas]].
Then $\FF$ can be extended to (that is, is a [[Definition:Subset|subset]] of) a [[Definition:Maximal Consistent Set of Formulas|maximal consistent set of formulas... | {{ProofWanted|This is probably the same as [[Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension]]}} | Consistent Set of Formulas can be Extended to Maximal Consistent Set | https://proofwiki.org/wiki/Consistent_Set_of_Formulas_can_be_Extended_to_Maximal_Consistent_Set | https://proofwiki.org/wiki/Consistent_Set_of_Formulas_can_be_Extended_to_Maximal_Consistent_Set | [
"Logical Consistency"
] | [
"Definition:Collection",
"Definition:Consistent (Logic)/Set of Formulas",
"Definition:Logical Formula",
"Definition:Subset",
"Definition:Maximal Consistent Set of Formulas"
] | [
"Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension"
] |
proofwiki-19604 | Tukey's Lemma/Formulation 1 | Let $S$ be a non-empty set of finite character.
Then $S$ has an element which is maximal with respect to the subset relation. | Let $C \subseteq S$ be a chain.
We will show that $\ds \bigcup C \in S$.
Let $x$ be a finite subset of $\ds \bigcup C$.
By the definitions of subset and of union, each element of $x$ is an element of at least one element of $C$.
By the Principle of Finite Choice, there is a mapping $c: x \to C$ such that:
:$\forall a \... | Let $S$ be a [[Definition:Non-Empty Set|non-empty set]] of [[Definition:Finite Character|finite character]].
Then $S$ has an [[Definition:Element|element]] which is [[Definition:Maximal Element|maximal]] with respect to the [[Definition:Subset Relation|subset relation]]. | Let $C \subseteq S$ be a [[Definition:Chain of Sets|chain]].
We will show that $\ds \bigcup C \in S$.
Let $x$ be a [[Definition:Finite Subset|finite subset]] of $\ds \bigcup C$.
By the definitions of [[Definition:Subset|subset]] and of [[Definition:Set Union|union]], each [[Definition:Element|element]] of $x$ is an ... | Tukey's Lemma/Formulation 1 | https://proofwiki.org/wiki/Tukey's_Lemma/Formulation_1 | https://proofwiki.org/wiki/Tukey's_Lemma/Formulation_1 | [
"Tukey's Lemma"
] | [
"Definition:Non-Empty Set",
"Definition:Finite Character",
"Definition:Element",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Finite Subset",
"Definition:Subset",
"Definition:Set Union",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Principle of Finite Choice",
"Definition:Mapping",
"Definition:Finite Subset",
"Finite Totally Ordered Se... |
proofwiki-19605 | Change of Measures Formula for Integrals | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\struct {X, \Sigma}$ such that:
:$\nu$ is absolutely continuous with respect to $\mu$.
Let $g$ be a Radon-Nikodym derivative of $\nu$ with respect to $\mu$.
Let $f : X \to \overline \R$ be a positive $\Sigma$-measurabl... | First consider $f$ a positive simple function.
From Simple Function has Standard Representation, there exists:
:a finite sequence $a_0, \ldots, a_n$ of non-negative real numbers
:a partition $E_0, E_1, \ldots, E_n$ of $X$ into $\Sigma$-measurable sets
such that:
:$\ds f = \sum_{i \mathop = 0}^n a_i \chi_{E_i}$
Then:
... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ and $\nu$ be [[Definition:Sigma-Finite Measure|$\sigma$-finite measures]] on $\struct {X, \Sigma}$ such that:
:$\nu$ is [[Definition:Absolutely Continuous Measure|absolutely continuous]] with respect to $\mu$.
Let $g$ be a [[... | First consider $f$ a [[Definition:Positive Simple Function|positive simple function]].
From [[Simple Function has Standard Representation]], there exists:
:a [[Definition:Finite Sequence|finite sequence]] $a_0, \ldots, a_n$ of [[Definition:Non-Negative Real Number|non-negative real numbers]]
:a [[Definition:Partitio... | Change of Measures Formula for Integrals | https://proofwiki.org/wiki/Change_of_Measures_Formula_for_Integrals | https://proofwiki.org/wiki/Change_of_Measures_Formula_for_Integrals | [
"Radon-Nikodym Derivatives",
"Change of Measures Formula for Integrals"
] | [
"Definition:Measurable Space",
"Definition:Sigma-Finite Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Radon-Nikodym Derivative",
"Definition:Measurable Function/Positive",
"Definition:Pointwise Multiplication of Extended Real-Valued Functions",
"Definition:Integral of Positive Measurab... | [
"Definition:Simple Function",
"Measurable Function is Simple Function iff Finite Image Set/Corollary",
"Definition:Finite Sequence",
"Definition:Positive/Real Number",
"Definition:Set Partition",
"Definition:Measurable Set",
"Integral of Positive Measurable Function is Additive",
"Integral of Positive... |
proofwiki-19606 | Positive Part of Pointwise Product of Functions | Let $X$ be a set.
Let $f, g : X \to \overline \R$ be extended real-valued functions.
Then:
:$\map {\paren {f \cdot g}^+} x = \begin{cases}\map {f^+} x \map {g^+} x & \map f x \ge 0 \text { and } \map g x \ge 0 \\ \map {f^-} x \map {g^-} x & \map f x \le 0 \text { and } \map g x \le 0 \\ 0 & \text {otherwise}\end{case... | Let $x \in X$ be such that $\map f x \ge 0$ and $\map g x \ge 0$.
Then:
:$\map {\paren {f \cdot g} } x = \map f x \map g x \ge 0$
From the definition of the positive part, we then have:
:$\map {\paren {f \cdot g}^+} x = \map f x \map g x$
Now let $x \in X$ be such that $\map f x \le 0$ and $\map g x \le 0$.
Then, we ... | Let $X$ be a [[Definition:Set|set]].
Let $f, g : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|extended real-valued functions]].
Then:
:$\map {\paren {f \cdot g}^+} x = \begin{cases}\map {f^+} x \map {g^+} x & \map f x \ge 0 \text { and } \map g x \ge 0 \\ \map {f^-} x \map {g^-} x & \map f x ... | Let $x \in X$ be such that $\map f x \ge 0$ and $\map g x \ge 0$.
Then:
:$\map {\paren {f \cdot g} } x = \map f x \map g x \ge 0$
From the definition of the [[Definition:Positive Part|positive part]], we then have:
:$\map {\paren {f \cdot g}^+} x = \map f x \map g x$
Now let $x \in X$ be such that $\map f x \le ... | Positive Part of Pointwise Product of Functions | https://proofwiki.org/wiki/Positive_Part_of_Pointwise_Product_of_Functions | https://proofwiki.org/wiki/Positive_Part_of_Pointwise_Product_of_Functions | [
"Positive Part",
"Positive Parts",
"Positive Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Pointwise Multiplication of Extended Real-Valued Functions",
"Definition:Positive Part"
] | [
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Positive Part",
"Category:Positive Parts"
] |
proofwiki-19607 | Negative Part of Pointwise Product of Functions | Let $X$ be a set.
Let $f, g : X \to \overline \R$ be extended real-valued functions.
Then:
:$\paren {f \cdot g}^- = f^- g^+ + f^+ g^-$
where:
:$f \cdot g$ is the pointwise product of $f$ and $g$
:$\paren {f \cdot g}^-$ denotes the negative part. | We have:
{{begin-eqn}}
{{eqn | l = \map f x \map g x
| r = \paren {\map {f^+} x - \map {f^-} x} \paren {\map {g^+} x - \map {g^-} x}
| c = Difference of Positive and Negative Parts
}}
{{eqn | r = \map {f^+} x \map {g^+} x - \map {f^-} x \map {g^+} x - \map {f^+} x \map {g^-} x + \map {f^-} x \map {g^-} x
}}
{{end-... | Let $X$ be a [[Definition:Set|set]].
Let $f, g : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|extended real-valued functions]].
Then:
:$\paren {f \cdot g}^- = f^- g^+ + f^+ g^-$
where:
:$f \cdot g$ is the [[Definition:Pointwise Multiplication of Extended Real-Valued Functions|pointwise pro... | We have:
{{begin-eqn}}
{{eqn | l = \map f x \map g x
| r = \paren {\map {f^+} x - \map {f^-} x} \paren {\map {g^+} x - \map {g^-} x}
| c = [[Difference of Positive and Negative Parts]]
}}
{{eqn | r = \map {f^+} x \map {g^+} x - \map {f^-} x \map {g^+} x - \map {f^+} x \map {g^-} x + \map {f^-} x \map {g^-} x
}}
{... | Negative Part of Pointwise Product of Functions | https://proofwiki.org/wiki/Negative_Part_of_Pointwise_Product_of_Functions | https://proofwiki.org/wiki/Negative_Part_of_Pointwise_Product_of_Functions | [
"Negative Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Pointwise Multiplication of Extended Real-Valued Functions",
"Definition:Negative Part"
] | [
"Difference of Positive and Negative Parts",
"Category:Negative Parts"
] |
proofwiki-19608 | Swelled Class contains Empty Set | Let $A$ be a swelled class.
Then the empty set is an element of $A$. | By definition of swelled class, every subclass of every element of $A$ is also an element of $A$.
Let $x \in A$.
Then by Empty Class is Subclass of All Classes, the empty class is an element of $A$.
By the {{axiom-link|the Empty Set|Class Theory}}, the empty class is a set.
Hence the result.
{{qed}}
Category:Empty Set
... | Let $A$ be a [[Definition:Swelled Class|swelled class]].
Then the [[Definition:Empty Set|empty set]] is an [[Definition:Element of Class|element]] of $A$. | By definition of [[Definition:Swelled Class|swelled class]], every [[Definition:Subclass|subclass]] of every [[Definition:Element of Class|element]] of $A$ is also an [[Definition:Element of Class|element]] of $A$.
Let $x \in A$.
Then by [[Empty Class is Subclass of All Classes]], the [[Definition:Empty Class|empty c... | Swelled Class contains Empty Set | https://proofwiki.org/wiki/Swelled_Class_contains_Empty_Set | https://proofwiki.org/wiki/Swelled_Class_contains_Empty_Set | [
"Empty Set",
"Swelled Classes"
] | [
"Definition:Swelled Class",
"Definition:Empty Set",
"Definition:Element/Class"
] | [
"Definition:Swelled Class",
"Definition:Subclass",
"Definition:Element/Class",
"Definition:Element/Class",
"Empty Class is Subclass of All Classes",
"Definition:Empty Class",
"Definition:Element/Class",
"Definition:Empty Class",
"Definition:Set",
"Category:Empty Set",
"Category:Swelled Classes"
... |
proofwiki-19609 | Class of Finite Character is Swelled | Let $A$ be a class which has finite character.
Then $A$ is a swelled class. | Let $x \in A$ and $y \subseteq x$.
Then {{hypothesis}} every finite subset of $y$ is also a finite subset of $x$.
Hence every finite subset of $y$ is in $A$.
Hence again {{hypothesis}} $y \in A$.
{{qed}} | Let $A$ be a [[Definition:Class (Class Theory)|class]] which has [[Definition:Finite Character (Class Theory)|finite character]].
Then $A$ is a [[Definition:Swelled Class|swelled class]]. | Let $x \in A$ and $y \subseteq x$.
Then {{hypothesis}} every [[Definition:Finite Subset|finite subset]] of $y$ is also a [[Definition:Finite Subset|finite subset]] of $x$.
Hence every [[Definition:Finite Subset|finite subset]] of $y$ is in $A$.
Hence again {{hypothesis}} $y \in A$.
{{qed}} | Class of Finite Character is Swelled | https://proofwiki.org/wiki/Class_of_Finite_Character_is_Swelled | https://proofwiki.org/wiki/Class_of_Finite_Character_is_Swelled | [
"Swelled Classes",
"Finite Character"
] | [
"Definition:Class (Class Theory)",
"Definition:Finite Character/Class Theory",
"Definition:Swelled Class"
] | [
"Definition:Finite Subset",
"Definition:Finite Subset",
"Definition:Finite Subset"
] |
proofwiki-19610 | Class of Finite Character is Closed under Chain Unions | Let $A$ be a class which has finite character.
Then $A$ is closed under chain unions. | Let $C$ be a chain of elements of $A$.
$C$ is a set.
To show that $\bigcup C \in A$ it is sufficient to show that every finite subset of $\bigcup C$ is also an element of $A$.
Let $\set {y_1, y_2, \ldots, y_n}$ be an arbitrary subset of $\bigcup C$.
For each $i \le n$, we have that $y_i$ is an element of some $c_i$ of ... | Let $A$ be a [[Definition:Class (Class Theory)|class]] which has [[Definition:Finite Character (Class Theory)|finite character]].
Then $A$ is [[Definition:Closure under Chain Unions|closed under chain unions]]. | Let $C$ be a [[Definition:Chain of Sets|chain]] of [[Definition:Element of Class|elements]] of $A$.
$C$ is a [[Definition:Set|set]].
To show that $\bigcup C \in A$ it is sufficient to show that every [[Definition:Finite Subset|finite subset]] of $\bigcup C$ is also an [[Definition:Element of Class|element]] of $A$.
... | Class of Finite Character is Closed under Chain Unions | https://proofwiki.org/wiki/Class_of_Finite_Character_is_Closed_under_Chain_Unions | https://proofwiki.org/wiki/Class_of_Finite_Character_is_Closed_under_Chain_Unions | [
"Closure under Chain Unions",
"Finite Character"
] | [
"Definition:Class (Class Theory)",
"Definition:Finite Character/Class Theory",
"Definition:Closure under Chain Unions"
] | [
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Element/Class",
"Definition:Set",
"Definition:Finite Subset",
"Definition:Element/Class",
"Definition:Subset",
"Definition:Element",
"Definition:Greatest Element",
"Definition:Subset Relation",
"Definition:Set",
"Definition:Set",
"... |
proofwiki-19611 | Kuratowski's Lemma implies Tukey's Lemma | Let Kuratowski's Lemma be accepted as true.
Then Tukey's Lemma holds. | Recall Kuratowski's Lemma:
{{:Kuratowski's Lemma/Formulation 2}}{{qed|lemma}}
Recall Tukey's Lemma:
{{:Tukey's Lemma/Formulation 2}}{{qed|lemma}}
So, let us assume Kuratowski's Lemma.
Let $S$ be a non-empty set of finite character.
From Class of Finite Character is Closed under Chain Unions, $S$ is closed under chain u... | Let [[Kuratowski's Lemma]] be accepted as true.
Then [[Tukey's Lemma]] holds. | Recall [[Kuratowski's Lemma]]:
{{:Kuratowski's Lemma/Formulation 2}}{{qed|lemma}}
Recall [[Tukey's Lemma]]:
{{:Tukey's Lemma/Formulation 2}}{{qed|lemma}}
So, let us assume [[Kuratowski's Lemma]].
Let $S$ be a [[Definition:Non-Empty Set|non-empty set]] of [[Definition:Finite Character|finite character]].
From [[Cl... | Kuratowski's Lemma implies Tukey's Lemma | https://proofwiki.org/wiki/Kuratowski's_Lemma_implies_Tukey's_Lemma | https://proofwiki.org/wiki/Kuratowski's_Lemma_implies_Tukey's_Lemma | [
"Kuratowski's Lemma",
"Tukey's Lemma"
] | [
"Kuratowski's Lemma",
"Tukey's Lemma"
] | [
"Kuratowski's Lemma",
"Tukey's Lemma",
"Kuratowski's Lemma",
"Definition:Non-Empty Set",
"Definition:Finite Character",
"Class of Finite Character is Closed under Chain Unions",
"Definition:Closure under Chain Unions",
"Kuratowski's Lemma",
"Definition:Element",
"Definition:Subset",
"Definition:... |
proofwiki-19612 | Join of Finite Sub-Sigma-Algebras Generates Join of Finite Partitions | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\BB, \CC \subseteq \Sigma$ be finite sub-$\sigma$-algebras.
Then:
:$\map \xi {\BB \vee \CC} = \map \xi \BB \vee \map \xi \CC$
where:
:$\map\xi\cdot$ denotes the generated finite partition
:$\vee$ on the {{LHS}} denotes the join of finite sub-$\sigma$-alge... | {{proofWanted}} | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\BB, \CC \subseteq \Sigma$ be [[Definition:Finite Sub-Sigma-Algebra|finite sub-$\sigma$-algebras]].
Then:
:$\map \xi {\BB \vee \CC} = \map \xi \BB \vee \map \xi \CC$
where:
:$\map\xi\cdot$ denotes the [[Definition:Finit... | {{proofWanted}} | Join of Finite Sub-Sigma-Algebras Generates Join of Finite Partitions | https://proofwiki.org/wiki/Join_of_Finite_Sub-Sigma-Algebras_Generates_Join_of_Finite_Partitions | https://proofwiki.org/wiki/Join_of_Finite_Sub-Sigma-Algebras_Generates_Join_of_Finite_Partitions | [
"Probability Theory",
"Ergodic Theory"
] | [
"Definition:Probability Space",
"Definition:Finite Sub-Sigma-Algebra",
"Definition:Finite Partition Generated by Finite Sub-Sigma-Algebra",
"Definition:Join of Finite Sub-Sigma-Algebras",
"Definition:Join of Finite Partitions"
] | [] |
proofwiki-19613 | Axiom of Choice implies Tukey's Lemma | Let the Axiom of Choice be accepted.
Then Tukey's Lemma holds. | Let us assume the truth of the Axiom of Choice.
From Axiom of Choice implies Kuratowski's Lemma, it follows that Kuratowski's Lemma holds.
From Kuratowski's Lemma implies Tukey's Lemma, it follows that Tukey's Lemma holds.
Hence the result.
{{qed}} | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then [[Tukey's Lemma]] holds. | Let us assume the truth of the [[Axiom:Axiom of Choice|Axiom of Choice]].
From [[Axiom of Choice implies Kuratowski's Lemma]], it follows that [[Kuratowski's Lemma]] holds.
From [[Kuratowski's Lemma implies Tukey's Lemma]], it follows that [[Tukey's Lemma]] holds.
Hence the result.
{{qed}} | Axiom of Choice implies Tukey's Lemma | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Tukey's_Lemma | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Tukey's_Lemma | [
"Axiom of Choice",
"Tukey's Lemma"
] | [
"Axiom:Axiom of Choice",
"Tukey's Lemma"
] | [
"Axiom:Axiom of Choice",
"Axiom of Choice implies Kuratowski's Lemma",
"Kuratowski's Lemma",
"Kuratowski's Lemma implies Tukey's Lemma",
"Tukey's Lemma"
] |
proofwiki-19614 | Set of Finite Character with Choice Function is of Type M | Let $S$ be a set of sets with finite character.
Let there exist a choice function for $S$.
Then $S$ is of type $M$, that is:
:every element of $S$ is a subset of a maximal element of $S$ under the subset relation. | Let $S$ be according to the hypothesis.
From Class of Finite Character is Closed under Chain Unions:
:$S$ is closed under chain unions.
From Closed Set under Chain Unions with Choice Function is of Type $M$:
:$S$ is of type $M$.
{{qed}} | Let $S$ be a [[Definition:Set of Sets|set of sets]] with [[Definition:Finite Character (Class Theory)|finite character]].
Let there exist a [[Definition:Choice Function|choice function]] for $S$.
Then $S$ is of [[Definition:Type M Set|type $M$]], that is:
:every [[Definition:Element|element]] of $S$ is a [[Definition... | Let $S$ be according to the hypothesis.
From [[Class of Finite Character is Closed under Chain Unions]]:
:$S$ is [[Definition:Closure under Chain Unions|closed under chain unions]].
From [[Closed Set under Chain Unions with Choice Function is of Type M|Closed Set under Chain Unions with Choice Function is of Type $M$... | Set of Finite Character with Choice Function is of Type M | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Choice_Function_is_of_Type_M | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Choice_Function_is_of_Type_M | [
"Finite Character",
"Choice Functions",
"Type M Sets"
] | [
"Definition:Set of Sets",
"Definition:Finite Character/Class Theory",
"Definition:Choice Function",
"Definition:Type M Set",
"Definition:Element",
"Definition:Subset",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [
"Class of Finite Character is Closed under Chain Unions",
"Definition:Closure under Chain Unions",
"Closed Set under Chain Unions with Choice Function is of Type M",
"Definition:Type M Set"
] |
proofwiki-19615 | Set of Subsets of Finite Character of Countable Set is of Type M | Let $D$ be a countable set.
Let $S$ be a set of subsets of $D$ such that $S$ is of finite character.
Then $S$ is of type $M$, that is:
:every element of $S$ is a subset of a maximal element of $S$ under the subset relation. | We have that $\ds \bigcup S \subseteq D$.
Hence by Subset of Countable Set is Countable, $\ds \bigcup S$ is itself countable.
Hence from Countable Set has Choice Function, $S$ has a choice function.
The result follows from Set of Finite Character with Choice Function is of Type $M$.
{{qed}} | Let $D$ be a [[Definition:Countable Set|countable set]].
Let $S$ be a [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $D$ such that $S$ is of [[Definition:Finite Character|finite character]].
Then $S$ is of [[Definition:Type M Set|type $M$]], that is:
:every [[Definition:Element|element]] of $S$ is a [[De... | We have that $\ds \bigcup S \subseteq D$.
Hence by [[Subset of Countable Set is Countable]], $\ds \bigcup S$ is itself [[Definition:Countable Set|countable]].
Hence from [[Countable Set has Choice Function]], $S$ has a [[Definition:Choice Function|choice function]].
The result follows from [[Set of Finite Character ... | Set of Subsets of Finite Character of Countable Set is of Type M | https://proofwiki.org/wiki/Set_of_Subsets_of_Finite_Character_of_Countable_Set_is_of_Type_M | https://proofwiki.org/wiki/Set_of_Subsets_of_Finite_Character_of_Countable_Set_is_of_Type_M | [
"Finite Character",
"Choice Functions",
"Type M Sets",
"Countable Sets"
] | [
"Definition:Countable Set",
"Definition:Set",
"Definition:Subset",
"Definition:Finite Character",
"Definition:Type M Set",
"Definition:Element",
"Definition:Subset",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [
"Subset of Countable Set is Countable",
"Definition:Countable Set",
"Countable Set has Choice Function",
"Definition:Choice Function",
"Set of Finite Character with Choice Function is of Type M"
] |
proofwiki-19616 | Positive Part of Composition of Functions | Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces.
Let $T : X \to X'$ be a $\Sigma/\Sigma'$-measurable mapping.
Let $f : X' \to \overline \R$ be a function.
Then:
:$\paren {f \circ T}^+ = f^+ \circ T$
where $\paren {f \circ T}^+$ denotes the positive part of $f \circ T$. | Let $x \in X$ be such that $\map {\paren {f \circ T} } x = \map f {\map T x} \ge 0$.
Then $\map f {\map T x} = \map {f^+} {\map T x}$ by the definition of the positive part.
So:
{{begin-eqn}}
{{eqn | l = \map {\paren {f \circ T}^+} x
| r = \map {\paren {f \circ T} } x
| c = {{Defof|Positive Part}}
}}
{{eqn | r = \... | Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be [[Definition:Measurable Space|measurable spaces]].
Let $T : X \to X'$ be a [[Definition:Measurable Mapping|$\Sigma/\Sigma'$-measurable mapping]].
Let $f : X' \to \overline \R$ be a [[Definition:Function|function]].
Then:
:$\paren {f \circ T}^+ = f^+ \circ T... | Let $x \in X$ be such that $\map {\paren {f \circ T} } x = \map f {\map T x} \ge 0$.
Then $\map f {\map T x} = \map {f^+} {\map T x}$ by the definition of the [[Definition:Positive Part|positive part]].
So:
{{begin-eqn}}
{{eqn | l = \map {\paren {f \circ T}^+} x
| r = \map {\paren {f \circ T} } x
| c = {{Defof|... | Positive Part of Composition of Functions | https://proofwiki.org/wiki/Positive_Part_of_Composition_of_Functions | https://proofwiki.org/wiki/Positive_Part_of_Composition_of_Functions | [
"Positive Parts"
] | [
"Definition:Measurable Space",
"Definition:Measurable Mapping",
"Definition:Function",
"Definition:Positive Part"
] | [
"Definition:Positive Part",
"Definition:Positive Part",
"Category:Positive Parts"
] |
proofwiki-19617 | Negative Part of Composition of Functions | Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces.
Let $T : X \to X'$ be a $\Sigma/\Sigma'$-measurable mapping.
Let $f : X' \to \overline \R$ be a function.
Then:
:$\paren {f \circ T}^- = f^- \circ T$
where $\paren {f \circ T}^-$ denotes the positive part of $f \circ T$. | Let $x \in X$ be such that $\map {\paren {f \circ T} } x = \map f {\map T x} \le 0$.
Then $\map {f^-} {\map T x} = -\map f {\map T x}$ by the definition of the negative part.
So:
{{begin-eqn}}
{{eqn | l = \map {\paren {f \circ T}^-} x
| r = -\map {\paren {f \circ T} } x
| c = {{Defof|Negative Part}}
}}
{{eqn | r =... | Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be [[Definition:Measurable Space|measurable spaces]].
Let $T : X \to X'$ be a [[Definition:Measurable Mapping|$\Sigma/\Sigma'$-measurable mapping]].
Let $f : X' \to \overline \R$ be a [[Definition:Function|function]].
Then:
:$\paren {f \circ T}^- = f^- \circ T... | Let $x \in X$ be such that $\map {\paren {f \circ T} } x = \map f {\map T x} \le 0$.
Then $\map {f^-} {\map T x} = -\map f {\map T x}$ by the definition of the [[Definition:Negative Part|negative part]].
So:
{{begin-eqn}}
{{eqn | l = \map {\paren {f \circ T}^-} x
| r = -\map {\paren {f \circ T} } x
| c = {{Defo... | Negative Part of Composition of Functions | https://proofwiki.org/wiki/Negative_Part_of_Composition_of_Functions | https://proofwiki.org/wiki/Negative_Part_of_Composition_of_Functions | [
"Negative Parts"
] | [
"Definition:Measurable Space",
"Definition:Measurable Mapping",
"Definition:Function",
"Definition:Positive Part"
] | [
"Definition:Negative Part",
"Definition:Positive Part",
"Category:Negative Parts"
] |
proofwiki-19618 | Characteristic Function of Preimage | Let $X$ and $Y$ be sets.
Let $f : X \to Y$ be a function.
Let $A \subseteq Y$.
Then:
:$\chi_{f^{-1} \sqbrk A} = \chi_A \circ f$
where $\chi_{f^{-1} \sqbrk A}$ denotes the characteristic function of $f^{-1} \sqbrk A$. | We show that if $x \in f^{-1} \sqbrk A$, then:
:$\map {\paren {\chi_A \circ f} } x = 1$
and if $x \in X \setminus f^{-1} \sqbrk A$, then:
:$\map {\paren {\chi_A \circ f} } x = 0$
Let $x \in f^{-1} \sqbrk A$.
Then $\map f x \in A$, and so:
:$\map {\paren {\chi_A \circ f} } x = \map {\chi_A} {\map f x} = 1$
Now let $x... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f : X \to Y$ be a [[Definition:Function|function]].
Let $A \subseteq Y$.
Then:
:$\chi_{f^{-1} \sqbrk A} = \chi_A \circ f$
where $\chi_{f^{-1} \sqbrk A}$ denotes the [[Definition:Characteristic Function|characteristic function]] of $f^{-1} \sqbrk A$. | We show that if $x \in f^{-1} \sqbrk A$, then:
:$\map {\paren {\chi_A \circ f} } x = 1$
and if $x \in X \setminus f^{-1} \sqbrk A$, then:
:$\map {\paren {\chi_A \circ f} } x = 0$
Let $x \in f^{-1} \sqbrk A$.
Then $\map f x \in A$, and so:
:$\map {\paren {\chi_A \circ f} } x = \map {\chi_A} {\map f x} = 1$
No... | Characteristic Function of Preimage | https://proofwiki.org/wiki/Characteristic_Function_of_Preimage | https://proofwiki.org/wiki/Characteristic_Function_of_Preimage | [
"Characteristic Functions"
] | [
"Definition:Set",
"Definition:Function",
"Definition:Characteristic Function"
] | [
"Category:Characteristic Functions"
] |
proofwiki-19619 | Cauchy Sequence of Subring iff Cauchy Sequence of Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {S, \norm {\, \cdot \,}_S }$ be a normed division subring of $\struct {R, \norm {\, \cdot \,} }$.
Let $\sequence{x_n}$ be a sequence in $S$.
Then:
:$\sequence{x_n}$ is a Cauchy sequence in $\struct {S, \norm {\, \cdot \,}_S }$
{{iff}}:
:$\s... | The result follows immediately from:
:{{Defof|Cauchy Sequence (Normed Division Ring)}}
:{{Defof|Normed Division Subring}}
{{qed}}
Category:Normed Division Rings
kkccrrvgnse5m9bs91foaz42j7hkpxo | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {S, \norm {\, \cdot \,}_S }$ be a [[Definition:Normed Division Subring|normed division subring]] of $\struct {R, \norm {\, \cdot \,} }$.
Let $\sequence{x_n}$ be a [[Definition:Sequence|sequence]] in $S$... | The result follows immediately from:
:{{Defof|Cauchy Sequence (Normed Division Ring)}}
:{{Defof|Normed Division Subring}}
{{qed}}
[[Category:Normed Division Rings]]
kkccrrvgnse5m9bs91foaz42j7hkpxo | Cauchy Sequence of Subring iff Cauchy Sequence of Normed Division Ring | https://proofwiki.org/wiki/Cauchy_Sequence_of_Subring_iff_Cauchy_Sequence_of_Normed_Division_Ring | https://proofwiki.org/wiki/Cauchy_Sequence_of_Subring_iff_Cauchy_Sequence_of_Normed_Division_Ring | [
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Normed Division Subring",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Category:Normed Division Rings"
] |
proofwiki-19620 | Convergent Sequence of Continuous Real Functions is Integrable Termwise | Let $I = \closedint a b \subseteq \R$ be a closed real interval.
Let $\struct {\map CI, \norm {\, \cdot \,}_\infty}$ be the normed vector space of real-valued functions continuous on $I$ equipped with the supremum norm.
Let $\sequence {f_k}_{k \mathop \in \N_{>0} } \in C \closedint a b$ be a sequence.
Suppose $\ds \sum... | {{begin-eqn}}
{{eqn | l = \int_a^b \map f t \rd t
| r = T f
| c = {{Defof|Riemann Integral Operator}}
}}
{{eqn | r = T \lim_{n \mathop \to \infty} s_n
| c = {{Defof|Convergent Series/Normed Vector Space/Definition 2|Convergent Series in Normed Vector Space}}
}}
{{eqn | r = \lim_{n \mathop \to \infty} ... | Let $I = \closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\struct {\map CI, \norm {\, \cdot \,}_\infty}$ be the [[Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space|normed vector space]] of [[Definition:Space of Real... | {{begin-eqn}}
{{eqn | l = \int_a^b \map f t \rd t
| r = T f
| c = {{Defof|Riemann Integral Operator}}
}}
{{eqn | r = T \lim_{n \mathop \to \infty} s_n
| c = {{Defof|Convergent Series/Normed Vector Space/Definition 2|Convergent Series in Normed Vector Space}}
}}
{{eqn | r = \lim_{n \mathop \to \infty} ... | Convergent Sequence of Continuous Real Functions is Integrable Termwise | https://proofwiki.org/wiki/Convergent_Sequence_of_Continuous_Real_Functions_is_Integrable_Termwise | https://proofwiki.org/wiki/Convergent_Sequence_of_Continuous_Real_Functions_is_Integrable_Termwise | [
"Operator Theory",
"Definite Integrals",
"Linear Transformations",
"Continuous Transformations"
] | [
"Definition:Real Interval/Closed",
"Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Supremum Norm",
"Definition:Sequence",
"Definition:Convergent Series/Normed... | [
"Riemann Integral Operator is Continuous Linear Transformation",
"Continuous Mappings preserve Convergent Sequences"
] |
proofwiki-19621 | Tukey's Lemma implies Zorn's Lemma | Let Tukey's Lemma be accepted as true.
Then Zorn's Lemma holds. | Recall Tukey's Lemma:
{{:Tukey's Lemma/Formulation 2}}{{qed|lemma}}
Recall Zorn's Lemma:
{{:Zorn's Lemma/Formulation 2}}{{qed|lemma}}
So, let us assume Tukey's Lemma.
Let $S$ be a non-empty ordered set, with $T$ as defined.
From Property of being Totally Ordered is of Finite Character:
:$T$ is of finite character.
From... | Let [[Tukey's Lemma]] be accepted as true.
Then [[Zorn's Lemma]] holds. | Recall [[Tukey's Lemma]]:
{{:Tukey's Lemma/Formulation 2}}{{qed|lemma}}
Recall [[Zorn's Lemma]]:
{{:Zorn's Lemma/Formulation 2}}{{qed|lemma}}
So, let us assume [[Tukey's Lemma]].
Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Ordered Set|ordered set]], with $T$ as defined.
From [[Property of bei... | Tukey's Lemma implies Zorn's Lemma | https://proofwiki.org/wiki/Tukey's_Lemma_implies_Zorn's_Lemma | https://proofwiki.org/wiki/Tukey's_Lemma_implies_Zorn's_Lemma | [
"Tukey's Lemma",
"Zorn's Lemma"
] | [
"Tukey's Lemma",
"Zorn's Lemma"
] | [
"Tukey's Lemma",
"Zorn's Lemma",
"Tukey's Lemma",
"Definition:Non-Empty Set",
"Definition:Ordered Set",
"Property of being Totally Ordered is of Finite Character",
"Definition:Finite Character",
"Ordering on Singleton is Total Ordering",
"Definition:Non-Empty Set",
"Tukey's Lemma",
"Definition:E... |
proofwiki-19622 | Property of being Totally Ordered is of Finite Character | Let $P$ be the property of sets defined as:
:$\forall x: \map P x$ denotes that $x$ is totally ordered under a relation $\RR$.
Then $P$ is of finite character.
That is:
:$x$ is totally ordered under $\RR$
{{iff}}:
:every finite subset of $x$ is totally ordered under $\RR$. | === Sufficient Condition ===
Let $x$ be totally ordered under $\RR$.
Let $y \subseteq x$.
From Restriction of Total Ordering is Total Ordering it follows that $y$ is also a totally ordered under $\RR$.
This holds in particular if $y$ is a finite set.
Hence every finite subset of $x$ is totally ordered under $\RR$.
{{qe... | Let $P$ be the [[Definition:Property|property]] of [[Definition:Set|sets]] defined as:
:$\forall x: \map P x$ denotes that $x$ is [[Definition:Totally Ordered Set|totally ordered]] under a [[Definition:Relation|relation]] $\RR$.
Then $P$ is of [[Definition:Finite Character (Property of Sets)|finite character]].
That... | === Sufficient Condition ===
Let $x$ be [[Definition:Totally Ordered Set|totally ordered]] under $\RR$.
Let $y \subseteq x$.
From [[Restriction of Total Ordering is Total Ordering]] it follows that $y$ is also a [[Definition:Totally Ordered Set|totally ordered]] under $\RR$.
This holds in particular if $y$ is a [[D... | Property of being Totally Ordered is of Finite Character | https://proofwiki.org/wiki/Property_of_being_Totally_Ordered_is_of_Finite_Character | https://proofwiki.org/wiki/Property_of_being_Totally_Ordered_is_of_Finite_Character | [
"Total Orderings",
"Finite Character"
] | [
"Definition:Property",
"Definition:Set",
"Definition:Totally Ordered Set",
"Definition:Relation",
"Definition:Finite Character/Property of Sets",
"Definition:Totally Ordered Set",
"Definition:Finite Subset",
"Definition:Totally Ordered Set"
] | [
"Definition:Totally Ordered Set",
"Restriction of Total Ordering is Total Ordering",
"Definition:Totally Ordered Set",
"Definition:Finite Set",
"Definition:Finite Subset",
"Definition:Totally Ordered Set",
"Definition:Finite Subset",
"Definition:Totally Ordered Set",
"Definition:Totally Ordered Set"... |
proofwiki-19623 | Midpoint-Convex Function is Rational Convex | Let $I$ be a non-empty real interval.
Let $f: I \to \R$ be a real function.
If $f$ is midpoint-convex, then $f$ is rational-convex. | It suffices to show that for each $n \in \N$ and for any choice of $n$ elements $x_1, \dots, x_n \in I$:
{{begin-eqn}}
{{eqn | l = \map f {\frac {x_1 + \dots + x_n} n}
| o = \leq
| r = \frac {\map f {x_1} + \dots + \map f {x_n} } n
}}
{{end-eqn}}
via Forward-Backward Induction. | Let $I$ be a [[Definition:Real Interval|non-empty real interval]].
Let $f: I \to \R$ be a [[Definition:Real Function|real function]].
If $f$ is [[Definition:Midpoint-Convex|midpoint-convex]], then $f$ is [[Definition:Rational Convex|rational-convex]]. | It suffices to show that for each $n \in \N$ and for any choice of $n$ [[Definition:Element|elements]] $x_1, \dots, x_n \in I$:
{{begin-eqn}}
{{eqn | l = \map f {\frac {x_1 + \dots + x_n} n}
| o = \leq
| r = \frac {\map f {x_1} + \dots + \map f {x_n} } n
}}
{{end-eqn}}
via [[Forward-Backward Induction]]. | Midpoint-Convex Function is Rational Convex | https://proofwiki.org/wiki/Midpoint-Convex_Function_is_Rational_Convex | https://proofwiki.org/wiki/Midpoint-Convex_Function_is_Rational_Convex | [
"Real Analysis"
] | [
"Definition:Real Interval",
"Definition:Real Function",
"Definition:Midpoint-Convex",
"Definition:Rational Convex"
] | [
"Definition:Element",
"Forward-Backward Induction",
"Definition:Element"
] |
proofwiki-19624 | Condition for Existence of Expectation of Real-Valued Measurable Function composed with Absolutely Continuous Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the probability distribution of $X$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.
Let $h : \R \to \R$ be a $\map \BB \R$-measurable function.
Let ... | From Composition of Measurable Mappings is Measurable, we have:
:$\map h X$ is $\Sigma$-measurable.
So:
:$\map h X$ is a real-valued random variable.
From the definition of the probability distribution, we have:
:$P_X = X_\ast \Pr$
where $X_\ast \Pr$ denotes the pushforward measure of $\Pr$ under $X$.
Since $X$ is an... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Absolutely Continuous Random Variable|absolutely continuous random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the [[Definition:Probability Distribution|probability distribution]] o... | From [[Composition of Measurable Mappings is Measurable]], we have:
:$\map h X$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
So:
:$\map h X$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]].
From the definition of the [[Definition:Probability Distribution|probability distri... | Condition for Existence of Expectation of Real-Valued Measurable Function composed with Absolutely Continuous Random Variable | https://proofwiki.org/wiki/Condition_for_Existence_of_Expectation_of_Real-Valued_Measurable_Function_composed_with_Absolutely_Continuous_Random_Variable | https://proofwiki.org/wiki/Condition_for_Existence_of_Expectation_of_Real-Valued_Measurable_Function_composed_with_Absolutely_Continuous_Random_Variable | [
"Absolutely Continuous Random Variables",
"Expectation"
] | [
"Definition:Probability Space",
"Definition:Absolutely Continuous Random Variable",
"Definition:Probability Distribution",
"Definition:Borel Sigma-Algebra",
"Definition:Measurable Function",
"Definition:Lebesgue Measure",
"Definition:Integrable Random Variable",
"Definition:Integrable Function/Lebesgu... | [
"Composition of Measurable Mappings is Measurable",
"Definition:Measurable Function",
"Definition:Random Variable/Real-Valued",
"Definition:Probability Distribution",
"Definition:Pushforward Measure",
"Definition:Absolutely Continuous Random Variable",
"Definition:Measurable Function",
"Integral with ... |
proofwiki-19625 | Expectation of Real-Valued Measurable Function composed with Absolutely Continuous Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the probability distribution of $X$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.
Let $h : \R \to \R$ be a $\map \BB \R$-measurable function.
Let ... | From Composition of Measurable Mappings is Measurable:
:$\map h X$ is $\Sigma$-measurable.
So:
:$\map h X$ is a real-valued random variable.
From Characterization of Integrable Functions, we have that:
:$\map h X$ is integrable {{iff}} $\size {\map h X}$ is integrable.
We have:
{{begin-eqn}}
{{eqn | l = \int \size {\ma... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Absolutely Continuous Random Variable|absolutely continuous random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the [[Definition:Probability Distribution|probability distribution]] o... | From [[Composition of Measurable Mappings is Measurable]]:
:$\map h X$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
So:
:$\map h X$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]].
From [[Characterization of Integrable Functions]], we have that:
:$\map h X$ is [[Definition... | Expectation of Real-Valued Measurable Function composed with Absolutely Continuous Random Variable | https://proofwiki.org/wiki/Expectation_of_Real-Valued_Measurable_Function_composed_with_Absolutely_Continuous_Random_Variable | https://proofwiki.org/wiki/Expectation_of_Real-Valued_Measurable_Function_composed_with_Absolutely_Continuous_Random_Variable | [
"Expectation",
"Absolutely Continuous Random Variables"
] | [
"Definition:Probability Space",
"Definition:Absolutely Continuous Random Variable",
"Definition:Probability Distribution",
"Definition:Borel Sigma-Algebra",
"Definition:Measurable Function",
"Definition:Probability Density Function",
"Definition:Lebesgue Measure",
"Definition:Integrable Random Variabl... | [
"Composition of Measurable Mappings is Measurable",
"Definition:Measurable Function",
"Definition:Random Variable/Real-Valued",
"Characterization of Integrable Functions",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"Integral with respect to Pushforward Measure",
... |
proofwiki-19626 | Axiom of Choice implies Hausdorff's Maximal Principle | Let the Axiom of Choice be accepted.
Then Hausdorff's Maximal Principle holds. | Let $S$ be the set of all chains of $\PP$.
$S \ne \O$ since the empty set is an element of $S$.
From Subset Relation is Ordering, we have that $\struct {S, \subseteq}$ is partially ordered by inclusion.
Let $C$ be a totally ordered subset of $\struct {S, \subseteq}$.
Then $\bigcup C$ is a chain in $C$ by Set of Chains... | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then [[Hausdorff's Maximal Principle]] holds. | Let $S$ be the set of all [[Definition:Chain (Order Theory)|chains]] of $\PP$.
$S \ne \O$ since the [[Definition:Empty Set|empty set]] is an [[Definition:Element|element]] of $S$.
From [[Subset Relation is Ordering]], we have that $\struct {S, \subseteq}$ is [[Definition:Partial Ordering|partially ordered]] by inclu... | Axiom of Choice implies Hausdorff's Maximal Principle/Proof 1 | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Hausdorff's_Maximal_Principle | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Hausdorff's_Maximal_Principle/Proof_1 | [
"Hausdorff's Maximal Principle",
"Axiom of Choice",
"Axiom of Choice implies Hausdorff's Maximal Principle"
] | [
"Axiom:Axiom of Choice",
"Hausdorff's Maximal Principle"
] | [
"Definition:Chain (Order Theory)",
"Definition:Empty Set",
"Definition:Element",
"Subset Relation is Ordering",
"Definition:Partial Ordering",
"Symbols:Abbreviations/T/Toset",
"Definition:Chain (Order Theory)",
"Set of Chains is Closed under Chain Unions under Subset Relation",
"Definition:Subset",
... |
proofwiki-19627 | Axiom of Choice implies Hausdorff's Maximal Principle | Let the Axiom of Choice be accepted.
Then Hausdorff's Maximal Principle holds. | Let $\preceq$ be an ordering on the set $\PP$.
Let $X$ be a chain in $\struct {\PP, \preceq}$.
By definition, a maximal chain in $\PP$ that includes $X$ is a chain $Y$ in $\PP$ such that $X \subseteq Y$ and there is no chain $Z$ in $\PP$ with $X \subseteq Z$ and $Y \subsetneq Z$.
Let us define $\CC$ as:
:$\CC = \leftse... | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then [[Hausdorff's Maximal Principle]] holds. | Let $\preceq$ be an [[Definition:Ordered Set|ordering]] on the [[Definition:Set|set]] $\PP$.
Let $X$ be a [[Definition:Chain (Order Theory)|chain]] in $\struct {\PP, \preceq}$.
By definition, a [[Definition:Maximal Chain|maximal chain]] in $\PP$ that includes $X$ is a [[Definition:Chain (Order Theory)|chain]] $Y$ in ... | Axiom of Choice implies Hausdorff's Maximal Principle/Proof 2 | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Hausdorff's_Maximal_Principle | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Hausdorff's_Maximal_Principle/Proof_2 | [
"Hausdorff's Maximal Principle",
"Axiom of Choice",
"Axiom of Choice implies Hausdorff's Maximal Principle"
] | [
"Axiom:Axiom of Choice",
"Hausdorff's Maximal Principle"
] | [
"Definition:Ordered Set",
"Definition:Set",
"Definition:Chain (Order Theory)",
"Definition:Maximal Chain",
"Definition:Chain (Order Theory)",
"Definition:Chain (Order Theory)",
"Definition:Chain (Order Theory)",
"Definition:Maximal Chain",
"Definition:Maximal",
"Definition:Maximal",
"Zorn's Lemm... |
proofwiki-19628 | Axiom of Choice implies Hausdorff's Maximal Principle | Let the Axiom of Choice be accepted.
Then Hausdorff's Maximal Principle holds. | Let $\struct {\CC, \subseteq}$ be the set of all chains in $P$ ordered by inclusion.
By Set of Chains is Closed under Chain Unions under Subset Relation, $\CC$ is a chain complete ordered set.
Now define $f: \CC \to \CC$ as follows:
:If $C$ is a maximal chain then $\map f C = C$.
:Otherwise $f$ chooses arbitrarily, us... | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then [[Hausdorff's Maximal Principle]] holds. | Let $\struct {\CC, \subseteq}$ be the [[Definition:Set of Sets|set]] of all [[Definition:Chain (Order Theory)|chains]] in $P$ ordered by [[Definition:Subset|inclusion]].
By [[Set of Chains is Closed under Chain Unions under Subset Relation]], $\CC$ is a [[Definition:Chain Complete Set|chain complete]] [[Definition:Or... | Axiom of Choice implies Hausdorff's Maximal Principle/Proof 3 | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Hausdorff's_Maximal_Principle | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Hausdorff's_Maximal_Principle/Proof_3 | [
"Hausdorff's Maximal Principle",
"Axiom of Choice",
"Axiom of Choice implies Hausdorff's Maximal Principle"
] | [
"Axiom:Axiom of Choice",
"Hausdorff's Maximal Principle"
] | [
"Definition:Set of Sets",
"Definition:Chain (Order Theory)",
"Definition:Subset",
"Set of Chains is Closed under Chain Unions under Subset Relation",
"Definition:Inductive Ordered Set",
"Definition:Ordered Set",
"Definition:Maximal Chain",
"Axiom:Axiom of Choice",
"Definition:Chain (Order Theory)",
... |
proofwiki-19629 | Hausdorff's Maximal Principle implies Kuratowski's Lemma | Let Hausdorff's Maximal Principle be accepted as true.
Then Kuratowski's Lemma holds. | Recall Hausdorff's Maximal Principle:
{{:Hausdorff's Maximal Principle/Formulation 2}}{{qed|lemma}}
Recall Kuratowski's Lemma:
{{:Kuratowski's Lemma/Formulation 2}}{{qed|lemma}}
So, let us assume Hausdorff's Maximal Principle.
Let $S$ be a non-empty set of sets which is closed under chain unions.
$\set b$ is trivially ... | Let [[Hausdorff's Maximal Principle]] be accepted as true.
Then [[Kuratowski's Lemma]] holds. | Recall [[Hausdorff's Maximal Principle]]:
{{:Hausdorff's Maximal Principle/Formulation 2}}{{qed|lemma}}
Recall [[Kuratowski's Lemma]]:
{{:Kuratowski's Lemma/Formulation 2}}{{qed|lemma}}
So, let us assume [[Hausdorff's Maximal Principle]].
Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set of Sets... | Hausdorff's Maximal Principle implies Kuratowski's Lemma/Proof 1 | https://proofwiki.org/wiki/Hausdorff's_Maximal_Principle_implies_Kuratowski's_Lemma | https://proofwiki.org/wiki/Hausdorff's_Maximal_Principle_implies_Kuratowski's_Lemma/Proof_1 | [
"Hausdorff's Maximal Principle",
"Kuratowski's Lemma",
"Hausdorff's Maximal Principle implies Kuratowski's Lemma"
] | [
"Hausdorff's Maximal Principle",
"Kuratowski's Lemma"
] | [
"Hausdorff's Maximal Principle",
"Kuratowski's Lemma",
"Hausdorff's Maximal Principle",
"Definition:Non-Empty Set",
"Definition:Set of Sets",
"Definition:Closure under Chain Unions",
"Definition:Chain (Order Theory)/Subset Relation",
"Hausdorff's Maximal Principle",
"Definition:Subset",
"Definitio... |
proofwiki-19630 | Hausdorff's Maximal Principle implies Kuratowski's Lemma | Let Hausdorff's Maximal Principle be accepted as true.
Then Kuratowski's Lemma holds. | We have:
* Hausdorff's Maximal Principle implies Axiom of Choice
* Axiom of Choice implies Kuratowski's Lemma
{{qed}} | Let [[Hausdorff's Maximal Principle]] be accepted as true.
Then [[Kuratowski's Lemma]] holds. | We have:
* [[Hausdorff's Maximal Principle implies Axiom of Choice]]
* [[Axiom of Choice implies Kuratowski's Lemma]]
{{qed}} | Hausdorff's Maximal Principle implies Kuratowski's Lemma/Proof 2 | https://proofwiki.org/wiki/Hausdorff's_Maximal_Principle_implies_Kuratowski's_Lemma | https://proofwiki.org/wiki/Hausdorff's_Maximal_Principle_implies_Kuratowski's_Lemma/Proof_2 | [
"Hausdorff's Maximal Principle",
"Kuratowski's Lemma",
"Hausdorff's Maximal Principle implies Kuratowski's Lemma"
] | [
"Hausdorff's Maximal Principle",
"Kuratowski's Lemma"
] | [
"Hausdorff's Maximal Principle implies Axiom of Choice",
"Axiom of Choice implies Kuratowski's Lemma"
] |
proofwiki-19631 | Zorn's Lemma implies Hausdorff's Maximal Principle | Let Zorn's Lemma be accepted as true.
Then Hausdorff's Maximal Principle holds. | Recall Zorn's Lemma:
{{:Zorn's Lemma/Formulation 2}}{{qed|lemma}}
Recall Hausdorff's Maximal Principle:
{{:Hausdorff's Maximal Principle/Formulation 2}}{{qed|lemma}}
It is seen directly that Hausdorff's Maximal Principle is a special case of Zorn's Lemma where the ordering is the subset relation.
{{qed}} | Let [[Zorn's Lemma]] be accepted as true.
Then [[Hausdorff's Maximal Principle]] holds. | Recall [[Zorn's Lemma]]:
{{:Zorn's Lemma/Formulation 2}}{{qed|lemma}}
Recall [[Hausdorff's Maximal Principle]]:
{{:Hausdorff's Maximal Principle/Formulation 2}}{{qed|lemma}}
It is seen directly that [[Hausdorff's Maximal Principle]] is a special case of [[Zorn's Lemma]] where the [[Definition:Ordering|ordering]] is... | Zorn's Lemma implies Hausdorff's Maximal Principle | https://proofwiki.org/wiki/Zorn's_Lemma_implies_Hausdorff's_Maximal_Principle | https://proofwiki.org/wiki/Zorn's_Lemma_implies_Hausdorff's_Maximal_Principle | [
"Zorn's Lemma",
"Hausdorff's Maximal Principle"
] | [
"Zorn's Lemma",
"Hausdorff's Maximal Principle"
] | [
"Zorn's Lemma",
"Hausdorff's Maximal Principle",
"Hausdorff's Maximal Principle",
"Zorn's Lemma",
"Definition:Ordering",
"Definition:Subset Relation"
] |
proofwiki-19632 | Trace Sigma-Algebra of Measurable Set | Let $\struct {X, \Sigma}$ be a measurable space.
Let $A \in \Sigma$.
Let $\Sigma_A$ be the trace $\sigma$-algebra of $A$ in $\Sigma$.
Then:
:$\Sigma_A = \set {B \in \Sigma : B \subseteq A}$
That is, the elements of $\Sigma_A$ are precisely the $\Sigma$-measurable sets that are subsets of $A$. | Let:
:$\Sigma' = \set {B \in \Sigma : B \subseteq A}$
We first show that:
:$\Sigma_A \subseteq \Sigma'$.
Let $B \in \Sigma_A$, then there exists $S \in \Sigma$ such that:
:$B = A \cap S$
From Sigma-Algebra Closed under Finite Intersection, we have:
:$B \in \Sigma$
while, from Intersection is Subset, we have:
:$A \c... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $A \in \Sigma$.
Let $\Sigma_A$ be the [[Definition:Trace Sigma-Algebra|trace $\sigma$-algebra of $A$ in $\Sigma$]].
Then:
:$\Sigma_A = \set {B \in \Sigma : B \subseteq A}$
That is, the elements of $\Sigma_A$ are precisely the [... | Let:
:$\Sigma' = \set {B \in \Sigma : B \subseteq A}$
We first show that:
:$\Sigma_A \subseteq \Sigma'$.
Let $B \in \Sigma_A$, then there exists $S \in \Sigma$ such that:
:$B = A \cap S$
From [[Sigma-Algebra Closed under Finite Intersection]], we have:
:$B \in \Sigma$
while, from [[Intersection is Subset]], w... | Trace Sigma-Algebra of Measurable Set | https://proofwiki.org/wiki/Trace_Sigma-Algebra_of_Measurable_Set | https://proofwiki.org/wiki/Trace_Sigma-Algebra_of_Measurable_Set | [
"Trace Sigma-Algebras"
] | [
"Definition:Measurable Space",
"Definition:Trace Sigma-Algebra",
"Definition:Measurable Set",
"Definition:Subset"
] | [
"Sigma-Algebra Closed under Finite Intersection",
"Intersection is Subset",
"Definition:Subset",
"Intersection with Subset is Subset",
"Definition:Trace Sigma-Algebra",
"Definition:Subset",
"Category:Trace Sigma-Algebras"
] |
proofwiki-19633 | Maximal Principles are Equivalent | The Maximal Principles are equivalent:
* Kuratowski's Lemma
* Tukey's Lemma
* Zorn's Lemma
* Hausdorff's Maximal Principle | We have:
* Kuratowski's Lemma implies Tukey's Lemma
* Tukey's Lemma implies Zorn's Lemma
* Zorn's Lemma implies Hausdorff's Maximal Principle
* Hausdorff's Maximal Principle implies Kuratowski's Lemma
{{qed}} | The [[Maximal Principles]] are equivalent:
* [[Kuratowski's Lemma]]
* [[Tukey's Lemma]]
* [[Zorn's Lemma]]
* [[Hausdorff's Maximal Principle]] | We have:
* [[Kuratowski's Lemma implies Tukey's Lemma]]
* [[Tukey's Lemma implies Zorn's Lemma]]
* [[Zorn's Lemma implies Hausdorff's Maximal Principle]]
* [[Hausdorff's Maximal Principle implies Kuratowski's Lemma]]
{{qed}} | Maximal Principles are Equivalent | https://proofwiki.org/wiki/Maximal_Principles_are_Equivalent | https://proofwiki.org/wiki/Maximal_Principles_are_Equivalent | [
"Maximal Principles"
] | [
"Maximal Principles",
"Kuratowski's Lemma",
"Tukey's Lemma",
"Zorn's Lemma",
"Hausdorff's Maximal Principle"
] | [
"Kuratowski's Lemma implies Tukey's Lemma",
"Tukey's Lemma implies Zorn's Lemma",
"Zorn's Lemma implies Hausdorff's Maximal Principle",
"Hausdorff's Maximal Principle implies Kuratowski's Lemma"
] |
proofwiki-19634 | Axiom of Choice implies Maximal Principles | Let the Axiom of Choice be accepted.
Then the Maximal Principles hold. | From Maximal Principles are Equivalent, it is sufficient to demonstrate that any one of them is implied by the Axiom of Choice.
Indeed, we have several such theorems:
* Axiom of Choice implies Kuratowski's Lemma
* Axiom of Choice implies Tukey's Lemma
* Axiom of Choice implies Zorn's Lemma
* Axiom of Choice implies Hau... | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then the [[Maximal Principles]] hold. | From [[Maximal Principles are Equivalent]], it is sufficient to demonstrate that any one of them is implied by the [[Axiom:Axiom of Choice|Axiom of Choice]].
Indeed, we have several such theorems:
* [[Axiom of Choice implies Kuratowski's Lemma]]
* [[Axiom of Choice implies Tukey's Lemma]]
* [[Axiom of Choice implies... | Axiom of Choice implies Maximal Principles | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Maximal_Principles | https://proofwiki.org/wiki/Axiom_of_Choice_implies_Maximal_Principles | [
"Maximal Principles",
"Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Maximal Principles"
] | [
"Maximal Principles are Equivalent",
"Axiom:Axiom of Choice",
"Axiom of Choice implies Kuratowski's Lemma",
"Axiom of Choice implies Tukey's Lemma",
"Axiom of Choice implies Zorn's Lemma",
"Axiom of Choice implies Hausdorff's Maximal Principle"
] |
proofwiki-19635 | Hausdorff's Maximal Principle implies Axiom of Choice/Lemma | $\Sigma$ is closed under chain unions. | Let $f$ and $g$ be choice functions.
Then:
:$f \subseteq g$
{{iff}}:
:$\Dom f \subseteq \Dom g$
:$\forall x \in \Dom f: \map f x = \map g x$
{{ProofWanted|etc.}} | $\Sigma$ is [[Definition:Closure under Chain Unions|closed under chain unions.]] | Let $f$ and $g$ be [[Definition:Choice Function|choice functions]].
Then:
:$f \subseteq g$
{{iff}}:
:$\Dom f \subseteq \Dom g$
:$\forall x \in \Dom f: \map f x = \map g x$
{{ProofWanted|etc.}} | Hausdorff's Maximal Principle implies Axiom of Choice/Lemma | https://proofwiki.org/wiki/Hausdorff's_Maximal_Principle_implies_Axiom_of_Choice/Lemma | https://proofwiki.org/wiki/Hausdorff's_Maximal_Principle_implies_Axiom_of_Choice/Lemma | [
"Hausdorff's Maximal Principle implies Axiom of Choice"
] | [
"Definition:Closure under Chain Unions"
] | [
"Definition:Choice Function"
] |
proofwiki-19636 | Integral with respect to Restriction of Measure to Trace Sigma-Algebra of Measurable Set | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $A \in \Sigma$.
Let $\Sigma_A$ be the trace $\sigma$-algebra of $A$ in $\Sigma$.
Let $\mu \restriction_{\Sigma_A}$ be the restriction of $\mu$ to $\Sigma_A$.
Let $f : X \to \overline \R$ be a positive measurable function.
Let $f \restriction_A$ be the restriction ... | From Restriction of Measurable Function is Measurable on Trace Sigma-Algebra, $f \restriction_A$ is $\Sigma_A$-measurable.
Consider the case that $f$ is a positive simple function.
From Simple Function has Standard Representation, there exists:
:a finite sequence $a_0, \ldots, a_n$ of non-negative real numbers
:a part... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $A \in \Sigma$.
Let $\Sigma_A$ be the [[Definition:Trace Sigma-Algebra|trace $\sigma$-algebra of $A$ in $\Sigma$]].
Let $\mu \restriction_{\Sigma_A}$ be the [[Definition:Restriction of Measure to Trace Sigma-Algebra of Measurable Set... | From [[Restriction of Measurable Function is Measurable on Trace Sigma-Algebra]], $f \restriction_A$ is [[Definition:Measurable Function|$\Sigma_A$-measurable]].
Consider the case that $f$ is a [[Definition:Positive Simple Function|positive simple function]].
From [[Simple Function has Standard Representation]], ther... | Integral with respect to Restriction of Measure to Trace Sigma-Algebra of Measurable Set | https://proofwiki.org/wiki/Integral_with_respect_to_Restriction_of_Measure_to_Trace_Sigma-Algebra_of_Measurable_Set | https://proofwiki.org/wiki/Integral_with_respect_to_Restriction_of_Measure_to_Trace_Sigma-Algebra_of_Measurable_Set | [
"Trace Sigma-Algebras"
] | [
"Definition:Measure Space",
"Definition:Trace Sigma-Algebra",
"Definition:Restriction of Measure to Trace Sigma-Algebra of Measurable Set",
"Definition:Measurable Function/Positive",
"Definition:Restriction/Mapping"
] | [
"Restriction of Measurable Function is Measurable on Trace Sigma-Algebra",
"Definition:Measurable Function",
"Definition:Simple Function",
"Measurable Function is Simple Function iff Finite Image Set/Corollary",
"Definition:Finite Sequence",
"Definition:Positive/Real Number",
"Definition:Set Partition",... |
proofwiki-19637 | Integral of Probability Density Function over the Reals is Equal to One | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an absolutely continuous random variable.
Let $f_X$ be a probability density function for $X$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.
Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.
Then:
:$\ds \int_\R f_... | Let $P_X$ be the probability distribution of $X$.
From the definition of a probability density function, $f_X$ is a Radon-Nikodym derivative of $P_X$ with respect to $\lambda$.
We then have:
{{begin-eqn}}
{{eqn | l = 1
| r = \map {P_X} \R
| c = {{Defof|Probability Measure}}, Probability Distribution is Probability... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Absolutely Continuous Random Variable|absolutely continuous random variable]].
Let $f_X$ be a [[Definition:Probability Density Function|probability density function]] for $X$.
Let $\map \BB \R$ b... | Let $P_X$ be the [[Definition:Probability Distribution|probability distribution]] of $X$.
From the definition of a [[Definition:Probability Density Function|probability density function]], $f_X$ is a [[Definition:Radon-Nikodym Derivative|Radon-Nikodym derivative of $P_X$ with respect to $\lambda$]].
We then have:
{... | Integral of Probability Density Function over the Reals is Equal to One | https://proofwiki.org/wiki/Integral_of_Probability_Density_Function_over_the_Reals_is_Equal_to_One | https://proofwiki.org/wiki/Integral_of_Probability_Density_Function_over_the_Reals_is_Equal_to_One | [
"Probability Density Functions"
] | [
"Definition:Probability Space",
"Definition:Absolutely Continuous Random Variable",
"Definition:Probability Density Function",
"Definition:Borel Sigma-Algebra",
"Definition:Lebesgue Measure",
"Definition:Lebesgue Integral"
] | [
"Definition:Probability Distribution",
"Definition:Probability Density Function",
"Definition:Radon-Nikodym Derivative",
"Probability Distribution is Probability Measure",
"Category:Probability Density Functions"
] |
proofwiki-19638 | Equivalence of Definitions of Bounded Variation for Real Function on Closed Bounded Interval | Let $a, b$ be real numbers with $a < b$.
Let $f : \closedint a b \to \R$ be a real function.
{{TFAE|def = Bounded Variation (Closed Bounded Interval)}}
=== Definition 1 ===
{{:Definition:Bounded Variation/Closed Bounded Interval/Definition 1}}
=== Definition 2 ===
{{:Definition:Bounded Variation/Closed Bounded Interv... | === Definition 1 implies Definition 2 ===
Suppose that there exists a $M \in \R$ such that:
:$\map {V_f} {P ; \closedint a b} \le M$
for all finite subdivisions $P$.
Let $\SS$ be a finite non-empty subset of $\closedint a b$.
Define:
:$\SS^\ast = \SS \cup \set {a, b}$
and write:
:$\SS^\ast = \set {x_0, x_1, \ldots, ... | Let $a, b$ be [[Definition:Real Number|real numbers]] with $a < b$.
Let $f : \closedint a b \to \R$ be a [[Definition:Real Function|real function]].
{{TFAE|def = Bounded Variation (Closed Bounded Interval)}}
=== [[Definition:Bounded Variation/Closed Bounded Interval/Definition 1|Definition 1]] ===
{{:Definition:Bo... | === Definition 1 implies Definition 2 ===
Suppose that there exists a $M \in \R$ such that:
:$\map {V_f} {P ; \closedint a b} \le M$
for all [[Definition:Finite Subdivision|finite subdivisions]] $P$.
Let $\SS$ be a [[Definition:Finite Set|finite]] [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] ... | Equivalence of Definitions of Bounded Variation for Real Function on Closed Bounded Interval | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Bounded_Variation_for_Real_Function_on_Closed_Bounded_Interval | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Bounded_Variation_for_Real_Function_on_Closed_Bounded_Interval | [
"Bounded Variation"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Bounded Variation/Closed Bounded Interval/Definition 1",
"Definition:Bounded Variation/Closed Bounded Interval/Definition 2"
] | [
"Definition:Subdivision of Interval/Finite",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Subdivision of Interval/Finite",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Defi... |
proofwiki-19639 | Principle E is Equivalent to Kuratowski's Lemma | Kuratowski's Lemma is equivalent to Principle E. | Recall Kuratowski's Lemma:
{{:Kuratowski's Lemma/Formulation 2}}{{qed|lemma}}
Recall Principle E:
{{:Principle E}}{{qed|lemma}} | [[Kuratowski's Lemma]] is [[Definition:Logical Equivalence|equivalent]] to [[Principle E]]. | Recall [[Kuratowski's Lemma]]:
{{:Kuratowski's Lemma/Formulation 2}}{{qed|lemma}}
Recall [[Principle E]]:
{{:Principle E}}{{qed|lemma}} | Principle E is Equivalent to Kuratowski's Lemma | https://proofwiki.org/wiki/Principle_E_is_Equivalent_to_Kuratowski's_Lemma | https://proofwiki.org/wiki/Principle_E_is_Equivalent_to_Kuratowski's_Lemma | [
"Principle E",
"Kuratowski's Lemma"
] | [
"Kuratowski's Lemma",
"Definition:Logical Equivalence",
"Principle E"
] | [
"Kuratowski's Lemma",
"Principle E",
"Kuratowski's Lemma",
"Principle E",
"Principle E",
"Principle E",
"Principle E",
"Kuratowski's Lemma"
] |
proofwiki-19640 | Zorn's Lemma implies Kuratowski's Lemma | Let Zorn's Lemma be accepted as true.
Then Kuratowski's Lemma holds. | Recall Zorn's Lemma:
{{:Zorn's Lemma/Formulation 1}}{{qed|lemma}}
Recall Kuratowski's Lemma:
{{:Kuratowski's Lemma/Formulation 1}}{{qed|lemma}}
Let $\struct {S, \preceq}$ be a non-empty ordered set.
Let $C$ be a chain in $\struct {S, \preceq}$.
Let $T$ be the set of all chains in $\struct {S, \preceq}$ that are superse... | Let [[Zorn's Lemma]] be accepted as true.
Then [[Kuratowski's Lemma]] holds. | Recall [[Zorn's Lemma]]:
{{:Zorn's Lemma/Formulation 1}}{{qed|lemma}}
Recall [[Kuratowski's Lemma]]:
{{:Kuratowski's Lemma/Formulation 1}}{{qed|lemma}}
Let $\struct {S, \preceq}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Ordered Set|ordered set]].
Let $C$ be a [[Definition:Chain (Order Theory)|chain... | Zorn's Lemma implies Kuratowski's Lemma | https://proofwiki.org/wiki/Zorn's_Lemma_implies_Kuratowski's_Lemma | https://proofwiki.org/wiki/Zorn's_Lemma_implies_Kuratowski's_Lemma | [
"Zorn's Lemma",
"Kuratowski's Lemma"
] | [
"Zorn's Lemma",
"Kuratowski's Lemma"
] | [
"Zorn's Lemma",
"Kuratowski's Lemma",
"Definition:Non-Empty Set",
"Definition:Ordered Set",
"Definition:Chain (Order Theory)",
"Definition:Set",
"Definition:Chain (Order Theory)",
"Definition:Subset/Superset",
"Definition:Chain (Order Theory)",
"Definition:Power Set",
"Definition:Ordering",
"D... |
proofwiki-19641 | Bolzano-Weierstrass Theorem/Lemma 2 | Let $S$ be a non-empty subset of the real numbers such that its infimum $\map \inf s$ exists.
Let $\map \inf s \notin S$.
Then $\map \inf s$ is a limit point of $S$. | The proof follows exactly the same lines as Lemma $1$.
{{qed}} | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] such that its [[Definition:Infimum of Subset of Real Numbers|infimum]] $\map \inf s$ exists.
Let $\map \inf s \notin S$.
Then $\map \inf s$ is a [[Definition:Limit Point (Metric Space)|limit... | The proof follows exactly the same lines as [[Bolzano-Weierstrass Theorem/Lemma 1|Lemma $1$]].
{{qed}} | Bolzano-Weierstrass Theorem/Lemma 2 | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/Lemma_2 | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/Lemma_2 | [
"Bolzano-Weierstrass Theorem"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Infimum of Set/Real Numbers",
"Definition:Limit Point/Metric Space"
] | [
"Bolzano-Weierstrass Theorem/Lemma 1"
] |
proofwiki-19642 | Bolzano-Weierstrass Theorem/Lemma 3 | Every bounded, infinite subset $S$ of $\R$ has at least one limit point. | As $S$ is bounded, it is certainly bounded above.
Also, since {{hypothesis}} $S$ is infinite, it is of course non-empty.
Hence, by the completeness axiom of the real numbers, $\tilde s_0 = \sup S$ exists as a real.
There are two cases:
;Case $1.0$ -- $\tilde s_0 \notin S$:
By '''Lemma 1''', $\tilde s_0$ is a limit poin... | Every [[Definition:Bounded Subset of Real Numbers|bounded]], [[Definition:Infinite Set|infinite]] [[Definition:Subset|subset]] $S$ of $\R$ has at least one [[Definition:Limit Point (Metric Space)|limit point]]. | As $S$ is [[Definition:Bounded Subset of Real Numbers|bounded]], it is certainly [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Also, since {{hypothesis}} $S$ is [[Definition:Infinite Set|infinite]], it is of course [[Definition:Non-Empty Set|non-empty]].
Hence, by the [[Axiom:Real Number Axioms|c... | Bolzano-Weierstrass Theorem/Lemma 3 | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/Lemma_3 | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/Lemma_3 | [
"Bolzano-Weierstrass Theorem"
] | [
"Definition:Bounded Set/Real Numbers",
"Definition:Infinite Set",
"Definition:Subset",
"Definition:Limit Point/Metric Space"
] | [
"Definition:Bounded Set/Real Numbers",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Infinite Set",
"Definition:Non-Empty Set",
"Axiom:Real Number/Axioms",
"Definition:Real Number",
"Bolzano-Weierstrass Theorem/Lemma 1",
"Definition:Limit Point/Metric Space",
"Definition:Infinite Set",
... |
proofwiki-19643 | Restriction of Measure to Trace Sigma-Algebra of Measurable Set is Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $A \in \Sigma$.
Let $\Sigma_A$ be the trace $\sigma$-algebra of $A$ in $\Sigma$.
Let $\mu \restriction_{\Sigma_A}$ be the restriction of $\mu$ to $\Sigma_A$.
Then $\mu \restriction_{\Sigma_A}$ is a measure on $\struc... | We verify the three conditions required of a measure for $\mu \restriction_{\Sigma_A}$.
Note that from Trace Sigma-Algebra of Measurable Set, we have $\Sigma_A \subseteq \Sigma$. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Measure (Measure Theory)|measure]] on $\struct {X, \Sigma}$.
Let $A \in \Sigma$.
Let $\Sigma_A$ be the [[Definition:Trace Sigma-Algebra|trace $\sigma$-algebra of $A$ in $\Sigma$]].
Let $\mu \restriction_{\... | We verify the three conditions required of a [[Definition:Measure (Measure Theory)|measure]] for $\mu \restriction_{\Sigma_A}$.
Note that from [[Trace Sigma-Algebra of Measurable Set]], we have $\Sigma_A \subseteq \Sigma$. | Restriction of Measure to Trace Sigma-Algebra of Measurable Set is Measure | https://proofwiki.org/wiki/Restriction_of_Measure_to_Trace_Sigma-Algebra_of_Measurable_Set_is_Measure | https://proofwiki.org/wiki/Restriction_of_Measure_to_Trace_Sigma-Algebra_of_Measurable_Set_is_Measure | [
"Measures",
"Trace Sigma-Algebras"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Trace Sigma-Algebra",
"Definition:Restriction of Measure to Trace Sigma-Algebra of Measurable Set",
"Definition:Measure (Measure Theory)"
] | [
"Definition:Measure (Measure Theory)",
"Trace Sigma-Algebra of Measurable Set",
"Definition:Measure (Measure Theory)"
] |
proofwiki-19644 | Closure under Chain Unions with Choice Function implies Elements with no Immediate Extension | Let $S$ be a set of sets which:
:is closed under chain unions
:has a choice function $C$ for its union $\bigcup S$.
Let $b \in S$.
Then $b$ is the subset of an element of $S$ which has no immediate extension in $S$. | Let the hypothesis be assumed.
Let $A = \bigcup S$
Let $x \in S$ be arbitrary.
Let $\map E x$ be the set of all elements $a \in A$ such that $x \cup \set a$ is an immediate extension of $x$.
Note that:
:$\map E x \subseteq \bigcup S$
Suppose $x$ has an immediate extension in $S$.
Then:
:$\map E x$ is non-empty
:because... | Let $S$ be a [[Definition:Set of Sets|set of sets]] which:
:is [[Definition:Closure under Chain Unions|closed under chain unions]]
:has a [[Definition:Choice Function|choice function]] $C$ for its [[Definition:Union of Set of Sets|union]] $\bigcup S$.
Let $b \in S$.
Then $b$ is the [[Definition:Subset|subset]] of an ... | Let the hypothesis be assumed.
Let $A = \bigcup S$
Let $x \in S$ be arbitrary.
Let $\map E x$ be the [[Definition:Set|set]] of all [[Definition:Element|elements]] $a \in A$ such that $x \cup \set a$ is an [[Definition:Immediate Extension of Class|immediate extension]] of $x$.
Note that:
:$\map E x \subseteq \bigcup... | Closure under Chain Unions with Choice Function implies Elements with no Immediate Extension | https://proofwiki.org/wiki/Closure_under_Chain_Unions_with_Choice_Function_implies_Elements_with_no_Immediate_Extension | https://proofwiki.org/wiki/Closure_under_Chain_Unions_with_Choice_Function_implies_Elements_with_no_Immediate_Extension | [
"Closure under Chain Unions",
"Choice Functions"
] | [
"Definition:Set of Sets",
"Definition:Closure under Chain Unions",
"Definition:Choice Function",
"Definition:Set Union/Set of Sets",
"Definition:Subset",
"Definition:Element",
"Definition:Extension of Class/Immediate"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Extension of Class/Immediate",
"Definition:Extension of Class/Immediate",
"Definition:Non-Empty Set",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Choice Function",
"Definition:Progressing Mapping",
"Definition:Progressing Mapping",
"Def... |
proofwiki-19645 | Swelled Set which is Closed under Chain Unions with Choice Function is Type M | Let $S$ be a set of sets which:
:is closed under chain unions
:has a choice function $C$ for its union $\ds \bigcup S$.
Then:
:$S$ is swelled
{{iff}}:
:$S$ is of type $M$. | === Sufficient Condition ===
Let $S$ be swelled.
Let $b \in S$ be arbitrary.
From Closure under Chain Unions with Choice Function implies Elements with no Immediate Extension:
:$b$ is the subset of an element of $S$ which has no immediate extension in $S$.
Let $x \in S$ have no immediate extension in $S$.
Then from Ele... | Let $S$ be a [[Definition:Set of Sets|set of sets]] which:
:is [[Definition:Closure under Chain Unions|closed under chain unions]]
:has a [[Definition:Choice Function|choice function]] $C$ for its [[Definition:Union of Set of Sets|union]] $\ds \bigcup S$.
Then:
:$S$ is [[Definition:Swelled Class|swelled]]
{{iff}}:
:$... | === Sufficient Condition ===
Let $S$ be [[Definition:Swelled Class|swelled]].
Let $b \in S$ be arbitrary.
From [[Closure under Chain Unions with Choice Function implies Elements with no Immediate Extension]]:
:$b$ is the [[Definition:Subset|subset]] of an [[Definition:Element|element]] of $S$ which has no [[Definit... | Swelled Set which is Closed under Chain Unions with Choice Function is Type M | https://proofwiki.org/wiki/Swelled_Set_which_is_Closed_under_Chain_Unions_with_Choice_Function_is_Type_M | https://proofwiki.org/wiki/Swelled_Set_which_is_Closed_under_Chain_Unions_with_Choice_Function_is_Type_M | [
"Closure under Chain Unions",
"Choice Functions",
"Swelled Classes",
"Type M Sets"
] | [
"Definition:Set of Sets",
"Definition:Closure under Chain Unions",
"Definition:Choice Function",
"Definition:Set Union/Set of Sets",
"Definition:Swelled Class",
"Definition:Type M Set"
] | [
"Definition:Swelled Class",
"Closure under Chain Unions with Choice Function implies Elements with no Immediate Extension",
"Definition:Subset",
"Definition:Element",
"Definition:Extension of Class/Immediate",
"Definition:Extension of Class/Immediate",
"Element of Swelled Set with no Immediate Extension... |
proofwiki-19646 | Element of Swelled Set with no Immediate Extension is Maximal | Let $S$ be a swelled set of sets.
Let $x \in S$ have no immediate extension.
Then $x$ is a maximal element of $S$ {{WRT}} the subset relation. | {{ProofWanted|S&F merely say "We recall that ..."}} | Let $S$ be a [[Definition:Swelled Class|swelled]] [[Definition:Set of Sets|set of sets]].
Let $x \in S$ have no [[Definition:Immediate Extension of Class|immediate extension]].
Then $x$ is a [[Definition:Maximal Element|maximal element]] of $S$ {{WRT}} the [[Definition:Subset Relation|subset relation]]. | {{ProofWanted|S&F merely say "We recall that ..."}} | Element of Swelled Set with no Immediate Extension is Maximal | https://proofwiki.org/wiki/Element_of_Swelled_Set_with_no_Immediate_Extension_is_Maximal | https://proofwiki.org/wiki/Element_of_Swelled_Set_with_no_Immediate_Extension_is_Maximal | [
"Swelled Classes"
] | [
"Definition:Swelled Class",
"Definition:Set of Sets",
"Definition:Extension of Class/Immediate",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [] |
proofwiki-19647 | Equivalence of Definitions of Concentration of Signed Measure on Measurable Set | Let $\struct {X, \Sigma}$ be a measurable space.
{{TFAE|def = Concentration of Signed Measure on Measurable Set}} | From Characterization of Null Sets of Variation of Signed Measure, we have that:
:$\map {\size \mu} {E^c} = 0$ {{iff}}:
::for each $\Sigma$-measurable set $A \subseteq E^c$, we have $\map \mu A = 0$.
Hence the desired equivalence.
{{qed}}
Category:Concentration of Signed Measure on Measurable Set
hwp758lpy8w4m1pjdk50... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
{{TFAE|def = Concentration of Signed Measure on Measurable Set}} | From [[Characterization of Null Sets of Variation of Signed Measure]], we have that:
:$\map {\size \mu} {E^c} = 0$ {{iff}}:
::for each [[Definition:Measurable Set|$\Sigma$-measurable set]] $A \subseteq E^c$, we have $\map \mu A = 0$.
Hence the desired equivalence.
{{qed}}
[[Category:Concentration of Signed Measur... | Equivalence of Definitions of Concentration of Signed Measure on Measurable Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Concentration_of_Signed_Measure_on_Measurable_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Concentration_of_Signed_Measure_on_Measurable_Set | [
"Concentration of Signed Measure on Measurable Set"
] | [
"Definition:Measurable Space"
] | [
"Characterization of Null Sets of Variation of Signed Measure",
"Definition:Measurable Set",
"Category:Concentration of Signed Measure on Measurable Set"
] |
proofwiki-19648 | Equivalence of Definitions of Concentration of Complex Measure on Measurable Set | Let $\struct {X, \Sigma}$ be a measurable space.
{{TFAE|def = Concentration of Complex Measure on Measurable Set}} | From Characterization of Null Sets of Variation of Complex Measure, we have that:
:$\map {\size \mu} {E^c} = 0$ {{iff}}:
::for each $\Sigma$-measurable set $A \subseteq E^c$, we have $\map \mu A = 0$.
Hence the desired equivalence.
{{qed}}
Category:Concentration of Complex Measure on Measurable Set
pacgw5tun3r9qk5xue... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
{{TFAE|def = Concentration of Complex Measure on Measurable Set}} | From [[Characterization of Null Sets of Variation of Complex Measure]], we have that:
:$\map {\size \mu} {E^c} = 0$ {{iff}}:
::for each [[Definition:Measurable Set|$\Sigma$-measurable set]] $A \subseteq E^c$, we have $\map \mu A = 0$.
Hence the desired equivalence.
{{qed}}
[[Category:Concentration of Complex Meas... | Equivalence of Definitions of Concentration of Complex Measure on Measurable Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Concentration_of_Complex_Measure_on_Measurable_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Concentration_of_Complex_Measure_on_Measurable_Set | [
"Concentration of Complex Measure on Measurable Set"
] | [
"Definition:Measurable Space"
] | [
"Characterization of Null Sets of Variation of Complex Measure",
"Definition:Measurable Set",
"Category:Concentration of Complex Measure on Measurable Set"
] |
proofwiki-19649 | Lebesgue Decomposition Theorem/Finite Signed Measure | Let $\nu$ be a finite signed measure on $\struct {X, \Sigma}$.
Then there exist finite signed measures $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is absolutely continuous with respect to $\mu$
:$(2): \quad \nu_s$ and $\mu$ are mutually singular
:$(3): \quad \nu = \nu_a + \nu_s$ | This follows immediately from Finite Signed Measure is Complex Measure.
{{qed}} | Let $\nu$ be a [[Definition:Finite Signed Measure|finite signed measure]] on $\struct {X, \Sigma}$.
Then there exist [[Definition:Finite Signed Measure|finite signed measures]] $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is [[Definition:Absolutely Continuous Signed Measure|absolutel... | This follows immediately from [[Finite Signed Measure is Complex Measure]].
{{qed}} | Lebesgue Decomposition Theorem/Finite Signed Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Finite_Signed_Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Finite_Signed_Measure | [
"Lebesgue Decomposition Theorem",
"Signed Measures"
] | [
"Definition:Finite Measure/Signed Measure",
"Definition:Finite Measure/Signed Measure",
"Definition:Absolute Continuity/Signed Measure",
"Definition:Mutually Singular Measures"
] | [
"Finite Signed Measure is Complex Measure"
] |
proofwiki-19650 | Lebesgue Decomposition Theorem/Complex Measure | Let $\nu$ be a complex measure on $\struct {X, \Sigma}$.
Then there exists complex measures $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is absolutely continuous with respect to $\mu$
:$(2): \quad \nu_s$ and $\mu$ are mutually singular
:$(3): \quad \nu = \nu_a + \nu_s$ | Let $\cmod \nu$ be the variation of $\nu$.
From Variation of Complex Measure is Finite Measure, $\cmod \nu$ is a finite measure.
Then from Lebesgue Decomposition Theorem for Finite Measures, there exists finite measures $\cmod \nu_a$ and $\cmod \nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \cmod \nu_a$ is ab... | Let $\nu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Then there exists [[Definition:Complex Measure|complex measures]] $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is [[Definition:Absolute Continuity/Complex Measure|absolutely continuous]] with resp... | Let $\cmod \nu$ be the [[Definition:Variation of Complex Measure|variation]] of $\nu$.
From [[Variation of Complex Measure is Finite Measure]], $\cmod \nu$ is a [[Definition:Finite Measure|finite measure]].
Then from [[Lebesgue Decomposition Theorem for Finite Measures]], there exists [[Definition:Finite Measure|fini... | Lebesgue Decomposition Theorem/Complex Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Complex_Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Complex_Measure | [
"Lebesgue Decomposition Theorem",
"Complex Measures"
] | [
"Definition:Complex Measure",
"Definition:Complex Measure",
"Definition:Absolute Continuity/Complex Measure",
"Definition:Mutually Singular Measures"
] | [
"Definition:Variation/Complex Measure",
"Variation of Complex Measure is Finite Measure",
"Definition:Finite Measure",
"Lebesgue Decomposition Theorem for Finite Measures",
"Definition:Finite Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Mutually Singular Measures",
"Definition:Null ... |
proofwiki-19651 | Lebesgue Decomposition Theorem/Sigma-Finite Measure | Let $\nu$ be a $\sigma$-finite measure on $\struct {X, \Sigma}$.
Then there exists $\sigma$-finite measures $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is absolutely continuous with respect to $\mu$
:$(2): \quad \nu_s$ and $\mu$ are mutually singular
:$(3): \quad \nu = \nu_a + \nu_s$ | Since $\nu$ is $\sigma$-finite, there exists a sequence $\sequence {D_k}_{k \mathop \in \N}$ of disjoint sets $\Sigma$-measurable sets such that:
:$\map \nu {D_k} < \infty$
and:
:$\ds \bigcup_{k \mathop = 1}^\infty D_k = X$
For each $k \in \N$, introduce the intersection measures $\mu^{\paren k}$ and $\nu^{\paren k}$... | Let $\nu$ be a [[Definition:Sigma-Finite Measure|$\sigma$-finite measure]] on $\struct {X, \Sigma}$.
Then there exists [[Definition:Sigma-Finite Measure|$\sigma$-finite measures]] $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is [[Definition:Absolutely Continuous Measure|absolutely co... | Since $\nu$ is [[Definition:Sigma-Finite Measure|$\sigma$-finite]], there exists a [[Definition:Sequence|sequence]] $\sequence {D_k}_{k \mathop \in \N}$ of [[Definition:Disjoint Sets|disjoint sets]] [[Definition:Measurable Set|$\Sigma$-measurable sets]] such that:
:$\map \nu {D_k} < \infty$
and:
:$\ds \bigcup_{k \math... | Lebesgue Decomposition Theorem/Sigma-Finite Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Sigma-Finite_Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Sigma-Finite_Measure | [
"Lebesgue Decomposition Theorem",
"Sigma-Finite Measures"
] | [
"Definition:Sigma-Finite Measure",
"Definition:Sigma-Finite Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Mutually Singular Measures"
] | [
"Definition:Sigma-Finite Measure",
"Definition:Sequence",
"Definition:Disjoint Sets",
"Definition:Measurable Set",
"Definition:Intersection Measure",
"Definition:Trace Sigma-Algebra",
"Measure is Monotone",
"Definition:Finite Measure",
"Definition:Measurable Space",
"Lebesgue Decomposition Theorem... |
proofwiki-19652 | Lebesgue Decomposition Theorem/Finite Measure | Let $\nu$ be a finite measure on $\struct {X, \Sigma}$.
Then there exist finite measures $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is absolutely continuous with respect to $\mu$
:$(2): \quad \nu_s$ and $\mu$ are mutually singular
:$(3): \quad \nu = \nu_a + \nu_s$ | Define the set $\NN_\mu$ by:
:$\NN_\mu = \set {B \in \Sigma : \map \mu B = 0}$
Since $\nu$ is a finite measure, there exists $M \ge 0$ such that:
:$\map \nu A \le M$
for all $A \in \Sigma$.
So by the Continuum Property:
:the supremum of $\set {\map \nu B : B \in \NN_\mu}$ exists as a real number $L$.
By the definitio... | Let $\nu$ be a [[Definition:Finite Measure|finite measure]] on $\struct {X, \Sigma}$.
Then there exist [[Definition:Finite Measure|finite measures]] $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:
:$(1): \quad \nu_a$ is [[Definition:Absolutely Continuous Measure|absolutely continuous]] with respect to $\mu... | Define the set $\NN_\mu$ by:
:$\NN_\mu = \set {B \in \Sigma : \map \mu B = 0}$
Since $\nu$ is a [[Definition:Finite Measure|finite measure]], there exists $M \ge 0$ such that:
:$\map \nu A \le M$
for all $A \in \Sigma$.
So by the [[Continuum Property]]:
:the [[Definition:Supremum of Subset of Real Numbers|supre... | Lebesgue Decomposition Theorem/Finite Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Finite_Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Finite_Measure | [
"Lebesgue Decomposition Theorem",
"Measures"
] | [
"Definition:Finite Measure",
"Definition:Finite Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Mutually Singular Measures"
] | [
"Definition:Finite Measure",
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Squeeze Theorem",
"Null Sets Closed under Countable Union",
"Definition:Null Set",
"Set is Subset of Union",
"Definition:Supremum of S... |
proofwiki-19653 | Intersection Measure of Finite Measure is Finite Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a finite measure on $\struct {X, \Sigma}$.
Let $A \in \Sigma$.
Let $\mu_A$ be the intersection measure of $\mu$ by $A$.
Then $\mu_A$ is a finite measure. | From Intersection Measure is Measure, $\mu_A$ is a measure.
Since $\mu$ is a finite measure, we have:
:$\map \mu X < \infty$
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {\mu_A} X
| r = \map \mu {A \cap X}
| c = {{Defof|Intersection Measure}}
}}
{{eqn | r = \map \mu A
| c = Intersection with Subset is Subset
... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Finite Measure|finite measure]] on $\struct {X, \Sigma}$.
Let $A \in \Sigma$.
Let $\mu_A$ be the [[Definition:Intersection Measure|intersection measure of $\mu$ by $A$]].
Then $\mu_A$ is a [[Definition:Fi... | From [[Intersection Measure is Measure]], $\mu_A$ is a [[Definition:Measure (Measure Theory)|measure]].
Since $\mu$ is a [[Definition:Finite Measure|finite measure]], we have:
:$\map \mu X < \infty$
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {\mu_A} X
| r = \map \mu {A \cap X}
| c = {{Defof|Intersection M... | Intersection Measure of Finite Measure is Finite Measure | https://proofwiki.org/wiki/Intersection_Measure_of_Finite_Measure_is_Finite_Measure | https://proofwiki.org/wiki/Intersection_Measure_of_Finite_Measure_is_Finite_Measure | [
"Finite Measures",
"Intersection Measures"
] | [
"Definition:Measurable Space",
"Definition:Finite Measure",
"Definition:Intersection Measure",
"Definition:Finite Measure"
] | [
"Intersection Measure is Measure",
"Definition:Measure (Measure Theory)",
"Definition:Finite Measure",
"Intersection with Subset is Subset",
"Measure is Monotone",
"Definition:Finite Measure",
"Category:Finite Measures",
"Category:Intersection Measures"
] |
proofwiki-19654 | Intersection Complex Measure is Complex Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $F \in \Sigma$.
Let $\mu_F$ be the intersection complex measure of $\mu$ by $F$.
Then $\mu_F$ is a complex measure. | Since $\mu$ is a complex measure, we have:
:$\map \mu E \in \C$
for each $E \in \Sigma$.
So, in particular:
:$\map \mu {E \cap F} \in \C$
for all $E \in \Sigma$.
That is:
:$\map {\mu_F} E \in \C$
for all $E \in \Sigma$.
We verify the two conditions required of a complex measure.
We have:
{{begin-eqn}}
{{eqn | l = ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Let $F \in \Sigma$.
Let $\mu_F$ be the [[Definition:Intersection Complex Measure|intersection complex measure of $\mu$ by $F$]].
Then $\mu_F$ is ... | Since $\mu$ is a [[Definition:Complex Measure|complex measure]], we have:
:$\map \mu E \in \C$
for each $E \in \Sigma$.
So, in particular:
:$\map \mu {E \cap F} \in \C$
for all $E \in \Sigma$.
That is:
:$\map {\mu_F} E \in \C$
for all $E \in \Sigma$.
We verify the two conditions required of a [[Definition... | Intersection Complex Measure is Complex Measure | https://proofwiki.org/wiki/Intersection_Complex_Measure_is_Complex_Measure | https://proofwiki.org/wiki/Intersection_Complex_Measure_is_Complex_Measure | [
"Intersection Measures",
"Intersection Complex Measures",
"Complex Measures",
"Intersection Complex Measures"
] | [
"Definition:Measurable Space",
"Definition:Complex Measure",
"Definition:Intersection Measure/Complex Measure",
"Definition:Complex Measure"
] | [
"Definition:Complex Measure",
"Definition:Complex Measure",
"Intersection with Empty Set",
"Definition:Complex Measure",
"Definition:Sequence",
"Definition:Pairwise Disjoint",
"Intersection Distributes over Union",
"Definition:Countably Additive Function",
"Definition:Complex Measure",
"Category:C... |
proofwiki-19655 | Intersection Signed Measure is Signed Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $F \in \Sigma$.
Let $\mu_F$ be the intersection signed measure of $\mu$ by $F$.
Then $\mu_F$ is a signed measure. | Since $\mu$ is a signed measure it takes values in either $\overline \R \setminus \set \infty$ or $\overline \R \setminus \set {-\infty}$.
That is:
:$\map \mu E \in \overline \R \setminus \set \infty$ for each $E \in \Sigma$
or:
:$\map \mu E \in \overline \R \setminus \set {-\infty}$ for each $E \in \Sigma$.
In partic... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $F \in \Sigma$.
Let $\mu_F$ be the [[Definition:Intersection Signed Measure|intersection signed measure of $\mu$ by $F$]].
Then $\mu_F$ is a [[... | Since $\mu$ is a [[Definition:Signed Measure|signed measure]] it takes values in either $\overline \R \setminus \set \infty$ or $\overline \R \setminus \set {-\infty}$.
That is:
:$\map \mu E \in \overline \R \setminus \set \infty$ for each $E \in \Sigma$
or:
:$\map \mu E \in \overline \R \setminus \set {-\infty}$ ... | Intersection Signed Measure is Signed Measure | https://proofwiki.org/wiki/Intersection_Signed_Measure_is_Signed_Measure | https://proofwiki.org/wiki/Intersection_Signed_Measure_is_Signed_Measure | [
"Signed Measures",
"Intersection Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Intersection Measure/Signed Measure",
"Definition:Signed Measure"
] | [
"Definition:Signed Measure",
"Definition:Signed Measure",
"Intersection with Empty Set",
"Definition:Sequence",
"Definition:Pairwise Disjoint",
"Intersection Distributes over Union",
"Definition:Countably Additive Function",
"Definition:Signed Measure",
"Category:Signed Measures",
"Category:Inters... |
proofwiki-19656 | Set of Finite Character with Choice Function is Type M | Let $S$ be a set of sets of finite character.
Let $S$ have a choice function $C$ for its union $\ds \bigcup S$.
Then $S$ is of type $M$.
That is:
:every element of $S$ is a subset of a maximal element of $S$ under the subset relation. | By Class of Finite Character is Swelled, a set of finite character is swelled.
By Class of Finite Character is Closed under Chain Unions a set of finite character is closed under chain unions.
The result follows from Swelled Set which is Closed under Chain Unions with Choice Function is Type $M$.
{{qed}} | Let $S$ be a [[Definition:Set of Sets|set of sets]] of [[Definition:Finite Character|finite character]].
Let $S$ have a [[Definition:Choice Function|choice function]] $C$ for its [[Definition:Union of Set of Sets|union]] $\ds \bigcup S$.
Then $S$ is of [[Definition:Type M Set|type $M$]].
That is:
:every [[Definitio... | By [[Class of Finite Character is Swelled]], a [[Definition:Set|set]] of [[Definition:Finite Character|finite character]] is [[Definition:Swelled Class|swelled]].
By [[Class of Finite Character is Closed under Chain Unions]] a [[Definition:Set|set]] of [[Definition:Finite Character|finite character]] is [[Definition:... | Set of Finite Character with Choice Function is Type M | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Choice_Function_is_Type_M | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Choice_Function_is_Type_M | [
"Finite Character",
"Choice Functions",
"Type M Sets"
] | [
"Definition:Set of Sets",
"Definition:Finite Character",
"Definition:Choice Function",
"Definition:Set Union/Set of Sets",
"Definition:Type M Set",
"Definition:Element",
"Definition:Subset",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [
"Class of Finite Character is Swelled",
"Definition:Set",
"Definition:Finite Character",
"Definition:Swelled Class",
"Class of Finite Character is Closed under Chain Unions",
"Definition:Set",
"Definition:Finite Character",
"Definition:Closure under Chain Unions",
"Swelled Set which is Closed under ... |
proofwiki-19657 | Set of Finite Character with Countable Union is Type M | Let $S$ be a set of sets of finite character.
Let its union $\bigcup S$ be countable.
Then $S$ is of type $M$.
That is:
:every element of $S$ is a subset of a maximal element of $S$ under the subset relation. | By Countable Set has Choice Function, $S$ has a choice function.
The result follows from Set of Finite Character with Choice Function is Type $M$.
{{qed}} | Let $S$ be a [[Definition:Set of Sets|set of sets]] of [[Definition:Finite Character|finite character]].
Let its [[Definition:Union of Set of Sets|union]] $\bigcup S$ be [[Definition:Countable Set|countable]].
Then $S$ is of [[Definition:Type M Set|type $M$]].
That is:
:every [[Definition:Element|element]] of $S$ i... | By [[Countable Set has Choice Function]], $S$ has a [[Definition:Choice Function|choice function]].
The result follows from [[Set of Finite Character with Choice Function is Type M|Set of Finite Character with Choice Function is Type $M$]].
{{qed}} | Set of Finite Character with Countable Union is Type M/Proof 1 | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Countable_Union_is_Type_M | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Countable_Union_is_Type_M/Proof_1 | [
"Set of Finite Character with Countable Union is Type M",
"Finite Character",
"Choice Functions",
"Type M Sets"
] | [
"Definition:Set of Sets",
"Definition:Finite Character",
"Definition:Set Union/Set of Sets",
"Definition:Countable Set",
"Definition:Type M Set",
"Definition:Element",
"Definition:Subset",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [
"Countable Set has Choice Function",
"Definition:Choice Function",
"Set of Finite Character with Choice Function is Type M"
] |
proofwiki-19658 | Set of Finite Character with Countable Union is Type M | Let $S$ be a set of sets of finite character.
Let its union $\bigcup S$ be countable.
Then $S$ is of type $M$.
That is:
:every element of $S$ is a subset of a maximal element of $S$ under the subset relation. | Let $S$ be a set of sets of finite character whose union $\bigcup S$ is countable.
Let $D := \bigcup S$ be enumerated as:
:$D = \set {d_1, d_2, \ldots, d_n, d_{n + 1}, \ldots}$
Let $b \in S$ be arbitrary.
From the Principle of Recursive Definition, we can generate a countable sequence of elements of $S$ as follows:
Let... | Let $S$ be a [[Definition:Set of Sets|set of sets]] of [[Definition:Finite Character|finite character]].
Let its [[Definition:Union of Set of Sets|union]] $\bigcup S$ be [[Definition:Countable Set|countable]].
Then $S$ is of [[Definition:Type M Set|type $M$]].
That is:
:every [[Definition:Element|element]] of $S$ i... | Let $S$ be a [[Definition:Set of Sets|set of sets]] of [[Definition:Finite Character|finite character]] whose [[Definition:Union of Set of Sets|union]] $\bigcup S$ is [[Definition:Countable Set|countable]].
Let $D := \bigcup S$ be [[Definition:Enumeration|enumerated]] as:
:$D = \set {d_1, d_2, \ldots, d_n, d_{n + 1}, ... | Set of Finite Character with Countable Union is Type M/Proof 2 | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Countable_Union_is_Type_M | https://proofwiki.org/wiki/Set_of_Finite_Character_with_Countable_Union_is_Type_M/Proof_2 | [
"Set of Finite Character with Countable Union is Type M",
"Finite Character",
"Choice Functions",
"Type M Sets"
] | [
"Definition:Set of Sets",
"Definition:Finite Character",
"Definition:Set Union/Set of Sets",
"Definition:Countable Set",
"Definition:Type M Set",
"Definition:Element",
"Definition:Subset",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [
"Definition:Set of Sets",
"Definition:Finite Character",
"Definition:Set Union/Set of Sets",
"Definition:Countable Set",
"Definition:Enumeration",
"Principle of Recursive Definition",
"Definition:Countable Set",
"Definition:Sequence",
"Definition:Element",
"Definition:Set",
"Increasing Sequence ... |
proofwiki-19659 | Consistent Set of Logical Formulas is Subset of Maximally Consistent Set | Let $\LL$ be a formal language used in the field of symbolic logic.
Let $\FF$ be the set of logical formulas of $\LL$.
Let $\FF$ be countable.
Let $S$ be a consistent subset of $\FF$.
Then $S$ is a subset of some maximal consistent set of formulas. | {{ProofWanted|This may be the same as Consistent Set of Formulas can be Extended to Maximal Consistent Set. Smullyan and Fitting are vague here. They may be trying to state Lindenbaum's Lemma, but if this and that are equivalent is not clear.}} | Let $\LL$ be a [[Definition:Formal Language|formal language]] used in the field of [[Definition:Symbolic Logic|symbolic logic]].
Let $\FF$ be the [[Definition:Set|set]] of [[Definition:Logical Formula|logical formulas]] of $\LL$.
Let $\FF$ be [[Definition:Countable Set|countable]].
Let $S$ be a [[Definition:Consist... | {{ProofWanted|This may be the same as [[Consistent Set of Formulas can be Extended to Maximal Consistent Set]]. Smullyan and Fitting are vague here. They may be trying to state [[Lindenbaum's Lemma]], but if this and that are equivalent is not clear.}} | Consistent Set of Logical Formulas is Subset of Maximally Consistent Set | https://proofwiki.org/wiki/Consistent_Set_of_Logical_Formulas_is_Subset_of_Maximally_Consistent_Set | https://proofwiki.org/wiki/Consistent_Set_of_Logical_Formulas_is_Subset_of_Maximally_Consistent_Set | [
"Logical Consistency"
] | [
"Definition:Formal Language",
"Definition:Symbolic Logic",
"Definition:Set",
"Definition:Logical Formula",
"Definition:Countable Set",
"Definition:Consistent (Logic)/Set of Formulas",
"Definition:Subset",
"Definition:Subset",
"Definition:Maximal Consistent Set of Formulas"
] | [
"Consistent Set of Formulas can be Extended to Maximal Consistent Set",
"Lindenbaum's Lemma"
] |
proofwiki-19660 | Union of Chain in Set of Finite Character with Countable Union is Maximal Element | Let $S$ be a set of sets of finite character.
Let its union $\ds \bigcup S$ be countable.
Then $\ds \bigcup S$ is a maximal element of $S$ under the subset relation. | Let $S$ be as {{hypothesis}}.
{{AimForCont}} $\ds \bigcup S$ is not a maximal element of $S$ under the subset relation.
Then:
:$\exists T \subseteq S: \ds \bigcup S \subsetneqq T$
Thus:
:$\exists x \in S: x \ne \ds \bigcup S$
{{ProofWanted|need to ponder this, decide exactly what it's saying and what I can assume}} | Let $S$ be a [[Definition:Set of Sets|set of sets]] of [[Definition:Finite Character|finite character]].
Let its [[Definition:Union of Set of Sets|union]] $\ds \bigcup S$ be [[Definition:Countable Set|countable]].
Then $\ds \bigcup S$ is a [[Definition:Maximal Element|maximal element]] of $S$ under the [[Definition:... | Let $S$ be as {{hypothesis}}.
{{AimForCont}} $\ds \bigcup S$ is not a [[Definition:Maximal Element|maximal element]] of $S$ under the [[Definition:Subset Relation|subset relation]].
Then:
:$\exists T \subseteq S: \ds \bigcup S \subsetneqq T$
Thus:
:$\exists x \in S: x \ne \ds \bigcup S$
{{ProofWanted|need to ponder... | Union of Chain in Set of Finite Character with Countable Union is Maximal Element/Proof | https://proofwiki.org/wiki/Union_of_Chain_in_Set_of_Finite_Character_with_Countable_Union_is_Maximal_Element | https://proofwiki.org/wiki/Union_of_Chain_in_Set_of_Finite_Character_with_Countable_Union_is_Maximal_Element/Proof | [
"Union of Chain in Set of Finite Character with Countable Union is Maximal Element",
"Finite Character",
"Maximal Elements"
] | [
"Definition:Set of Sets",
"Definition:Finite Character",
"Definition:Set Union/Set of Sets",
"Definition:Countable Set",
"Definition:Maximal/Element",
"Definition:Subset Relation"
] | [
"Definition:Maximal/Element",
"Definition:Subset Relation"
] |
proofwiki-19661 | Finite Signed Measure is Complex Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a finite signed measure on $\struct {X, \Sigma}$.
Then $\mu$ is a complex measure on $\struct {X, \Sigma}$. | Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then:
:$\mu = \mu^+ - \mu^-$
for measures $\mu^+$ and $\mu^-$.
From Jordan Decomposition of Finite Signed Measure, $\mu^+$ and $\mu^-$ are finite measures.
Then, for each $A \in \Sigma$, we have:
{{begin-eqn}}
{{eqn | l = \cmod {\map \mu A}
| o = \le... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Finite Signed Measure|finite signed measure]] on $\struct {X, \Sigma}$.
Then $\mu$ is a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$. | Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then:
:$\mu = \mu^+ - \mu^-$
for [[Definition:Measure (Measure Theory)|measures]] $\mu^+$ and $\mu^-$.
From [[Jordan Decomposition of Finite Signed Measure]], $\mu^+$ and $\mu^-$ are [[Definition:Finite Measure|f... | Finite Signed Measure is Complex Measure | https://proofwiki.org/wiki/Finite_Signed_Measure_is_Complex_Measure | https://proofwiki.org/wiki/Finite_Signed_Measure_is_Complex_Measure | [
"Finite Signed Measures",
"Complex Measures"
] | [
"Definition:Measurable Space",
"Definition:Finite Measure/Signed Measure",
"Definition:Complex Measure"
] | [
"Definition:Jordan Decomposition",
"Definition:Measure (Measure Theory)",
"Jordan Decomposition of Finite Signed Measure",
"Definition:Finite Measure",
"Absolute Value of Signed Measure Bounded Above by Variation",
"Measure is Monotone",
"Definition:Real Number",
"Definition:Signed Measure",
"Defini... |
proofwiki-19662 | Distribution Function of Finite Signed Borel Measure is of Bounded Variation | Let $\mu$ be a finite signed Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then $F_\mu$ is of bounded variation. | Let $\SS$ be a non-empty finite subset of $\R$.
Write:
:$\SS = \set {x_0, x_1, \ldots, x_n}$
with:
:$x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n$
Then, we have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n \size {\map {F_\mu} {x_i} - \map {F_\mu} {x_{i - 1} } }
| r = \sum_{i \mathop = 1}^n \size {\map \mu {\hoi... | Let $\mu$ be a [[Definition:Finite Signed Measure|finite]] [[Definition:Signed Borel Measure|signed Borel measure]] on $\R$.
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Signed Borel Measure|distribution function]] of $\mu$.
Then $F_\mu$ is of [[Definition:Bounded Variation (Closed Unbounded Inte... | Let $\SS$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $\R$.
Write:
:$\SS = \set {x_0, x_1, \ldots, x_n}$
with:
:$x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n$
Then, we have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n \size {\map {F_\mu} {x_i}... | Distribution Function of Finite Signed Borel Measure is of Bounded Variation | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Signed_Borel_Measure_is_of_Bounded_Variation | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Signed_Borel_Measure_is_of_Bounded_Variation | [
"Distribution Function of Finite Signed Borel Measure",
"Bounded Variation"
] | [
"Definition:Finite Measure/Signed Measure",
"Definition:Signed Borel Measure",
"Definition:Distribution Function of Finite Signed Borel Measure",
"Definition:Bounded Variation/Closed Unbounded Interval"
] | [
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Measure of Set Difference with Subset/Signed Measure",
"Definition:Set Partition",
"Definition:Borel Sigma-Algebra/Borel Set",
"Definition:Variation/Signed Measure/Definition 2",
"Definition:Variation/Signed Measure",
"Signed... |
proofwiki-19663 | Measure of Set Difference with Subset/Signed Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $S, T \in \Sigma$ be such that $S \subseteq T$ with $\size {\map \mu S} < \infty$.
Then:
:$\map \mu {T \setminus S} = \map \mu T - \map \mu S$ | Let $\struct {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then, since:
:$\size {\map \mu S} < \infty$
We have:
:$\map {\mu^+} S < \infty$ and $\map {\mu^-} S < \infty$
Then, we have:
{{begin-eqn}}
{{eqn | l = \map \mu {T \setminus S}
| r = \map {\mu^+} {T \setminus S} - \map {\mu^-} {T \setminus S}
| c ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $S, T \in \Sigma$ be such that $S \subseteq T$ with $\size {\map \mu S} < \infty$.
Then:
:$\map \mu {T \setminus S} = \map \mu T - \map \mu S$ | Let $\struct {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then, since:
:$\size {\map \mu S} < \infty$
We have:
:$\map {\mu^+} S < \infty$ and $\map {\mu^-} S < \infty$
Then, we have:
{{begin-eqn}}
{{eqn | l = \map \mu {T \setminus S}
| r = \map {\mu^+} {T \setminu... | Measure of Set Difference with Subset/Signed Measure | https://proofwiki.org/wiki/Measure_of_Set_Difference_with_Subset/Signed_Measure | https://proofwiki.org/wiki/Measure_of_Set_Difference_with_Subset/Signed_Measure | [
"Measure of Set Difference with Subset",
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure"
] | [
"Definition:Jordan Decomposition",
"Measure of Set Difference with Subset",
"Category:Measure of Set Difference with Subset",
"Category:Signed Measures"
] |
proofwiki-19664 | Signed Measure Finite iff Finite Total Variation | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Then $\mu$ is finite {{iff}}:
:$\map {\size \mu} X < \infty$ | === Sufficient Condition ===
Suppose that:
:$\map {\size \mu} X < \infty$
Then, from Absolute Value of Signed Measure Bounded Above by Variation, we have:
:$\size {\map \mu X} \le \map {\size \mu} X$
so:
:$\size {\map \mu X} < \infty$
So $\mu$ is finite.
{{qed|lemma}} | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\size \mu$ be the [[Definition:Variation of Signed Measure|variation]] of $\mu$.
Then $\mu$ is [[Definition:Finite Signed Measure|finite]] {{i... | === Sufficient Condition ===
Suppose that:
:$\map {\size \mu} X < \infty$
Then, from [[Absolute Value of Signed Measure Bounded Above by Variation]], we have:
:$\size {\map \mu X} \le \map {\size \mu} X$
so:
:$\size {\map \mu X} < \infty$
So $\mu$ is [[Definition:Finite Signed Measure|finite]].
{{qed|lemma}} | Signed Measure Finite iff Finite Total Variation | https://proofwiki.org/wiki/Signed_Measure_Finite_iff_Finite_Total_Variation | https://proofwiki.org/wiki/Signed_Measure_Finite_iff_Finite_Total_Variation | [
"Signed Measures",
"Total Variation of Signed Measure"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Variation/Signed Measure",
"Definition:Finite Measure/Signed Measure"
] | [
"Absolute Value of Signed Measure Bounded Above by Variation",
"Definition:Finite Measure/Signed Measure",
"Definition:Finite Measure/Signed Measure"
] |
proofwiki-19665 | Measure of Limit of Decreasing Sequence of Measurable Sets | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an decreasing sequence of $\Sigma$-measurable sets such that:
:$E_n \downarrow E$
where $E_n \downarrow E$ denotes the limit of decreasing sequence of sets.
Suppose also that $\map \mu {E_1} < \infty$. ... | From Relative Complement of Decreasing Sequence of Sets is Increasing, we have:
:$\sequence {E_1 \setminus E_n}_{n \mathop \in \N}$ is increasing.
Further, we have:
{{begin-eqn}}
{{eqn | l = \bigcup_{n \mathop = 1}^\infty \paren {E_1 \setminus E_n}
| r = E_1 \setminus \paren {\bigcap_{n \mathop = 1}^\infty E_n}
|... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an [[Definition:Decreasing Sequence of Sets|decreasing sequence]] of [[Definition:Measurable Set|$\Sigma$-measurable sets]] such that:
:$E_n \downarrow E$
where $E_n \do... | From [[Relative Complement of Decreasing Sequence of Sets is Increasing]], we have:
:$\sequence {E_1 \setminus E_n}_{n \mathop \in \N}$ is [[Definition:Increasing Sequence of Sets|increasing]].
Further, we have:
{{begin-eqn}}
{{eqn | l = \bigcup_{n \mathop = 1}^\infty \paren {E_1 \setminus E_n}
| r = E_1 \setmin... | Measure of Limit of Decreasing Sequence of Measurable Sets | https://proofwiki.org/wiki/Measure_of_Limit_of_Decreasing_Sequence_of_Measurable_Sets | https://proofwiki.org/wiki/Measure_of_Limit_of_Decreasing_Sequence_of_Measurable_Sets | [
"Measures",
"Decreasing Sequences of Sets",
"Measure of Limit of Decreasing Sequence of Measurable Sets"
] | [
"Definition:Measure Space",
"Definition:Decreasing Sequence of Sets",
"Definition:Measurable Set",
"Definition:Limit of Decreasing Sequence of Sets"
] | [
"Relative Complement of Decreasing Sequence of Sets is Increasing",
"Definition:Increasing Sequence of Sets",
"De Morgan's Laws (Set Theory)/Set Difference/General Case",
"Measure of Limit of Increasing Sequence of Measurable Sets",
"Measure of Set Difference with Subset",
"Combination Theorem for Sequenc... |
proofwiki-19666 | Relative Complement of Decreasing Sequence of Sets is Increasing | Let $X$ be a set.
Let $\sequence {E_n}_{n \mathop \in \N}$ be a decreasing sequence of sets in $X$.
Then $\sequence {X \setminus E_n}_{n \mathop \in \N}$ is an increasing sequence of sets in $X$. | Since $\sequence {E_n}_{n \mathop \in \N}$ is decreasing, we have:
:$E_{n + 1} \subseteq E_n$ for each $n \in \N$.
From Relative Complement inverts Subsets, we then have:
:$X \setminus E_n \subseteq X \setminus E_{n + 1}$ for each $n \in \N$.
So $\sequence {X \setminus E_n}_{n \mathop \in \N}$ is an increasing sequ... | Let $X$ be a [[Definition:Set|set]].
Let $\sequence {E_n}_{n \mathop \in \N}$ be a [[Definition:Decreasing Sequence of Sets|decreasing sequence of sets]] in $X$.
Then $\sequence {X \setminus E_n}_{n \mathop \in \N}$ is an [[Definition:Increasing Sequence of Sets|increasing sequence of sets]] in $X$. | Since $\sequence {E_n}_{n \mathop \in \N}$ is [[Definition:Decreasing Sequence of Sets|decreasing]], we have:
:$E_{n + 1} \subseteq E_n$ for each $n \in \N$.
From [[Relative Complement inverts Subsets]], we then have:
:$X \setminus E_n \subseteq X \setminus E_{n + 1}$ for each $n \in \N$.
So $\sequence {X \setm... | Relative Complement of Decreasing Sequence of Sets is Increasing | https://proofwiki.org/wiki/Relative_Complement_of_Decreasing_Sequence_of_Sets_is_Increasing | https://proofwiki.org/wiki/Relative_Complement_of_Decreasing_Sequence_of_Sets_is_Increasing | [
"Decreasing Sequences of Sets",
"Increasing Sequences of Sets",
"Relative Complement",
"Decreasing Sequences of Sets"
] | [
"Definition:Set",
"Definition:Decreasing Sequence of Sets",
"Definition:Increasing Sequence of Sets"
] | [
"Definition:Decreasing Sequence of Sets",
"Relative Complement inverts Subsets",
"Definition:Increasing Sequence of Sets",
"Category:Increasing Sequences of Sets",
"Category:Relative Complement",
"Category:Decreasing Sequences of Sets"
] |
proofwiki-19667 | Limit of Tail of Decreasing Sequence of Sets | Let $X$ be a set.
Let $\sequence {E_n}_{n \mathop \in \N}$ be a decreasing sequence of subsets of $X$ such that:
:$E_n \downarrow E$
where $E_n \downarrow E$ denotes the limit of decreasing sequence of sets.
Then for each $m \in \N$ we have:
:$E_{n + m} \downarrow E$ | Let $m \in \N$.
From Tail of Decreasing Sequence of Sets is Decreasing, we have:
:$\sequence {E_{n + m} }_{n \mathop \in \N}$ is a decreasing sequence of sets.
Since:
:$E_n \downarrow E$
we have:
:$\ds \bigcap_{n \mathop = 1}^\infty E_n = E$
We show that:
:$\ds \bigcap_{n \mathop = 1}^\infty E_{n + m} = E$
That is:... | Let $X$ be a [[Definition:Set|set]].
Let $\sequence {E_n}_{n \mathop \in \N}$ be a [[Definition:Decreasing Sequence of Sets|decreasing sequence of subsets]] of $X$ such that:
:$E_n \downarrow E$
where $E_n \downarrow E$ denotes the [[Definition:Limit of Decreasing Sequence of Sets|limit of decreasing sequence of s... | Let $m \in \N$.
From [[Tail of Decreasing Sequence of Sets is Decreasing]], we have:
:$\sequence {E_{n + m} }_{n \mathop \in \N}$ is a [[Definition:Decreasing Sequence of Sets|decreasing sequence of sets]].
Since:
:$E_n \downarrow E$
we have:
:$\ds \bigcap_{n \mathop = 1}^\infty E_n = E$
We show that:
:$\ds... | Limit of Tail of Decreasing Sequence of Sets | https://proofwiki.org/wiki/Limit_of_Tail_of_Decreasing_Sequence_of_Sets | https://proofwiki.org/wiki/Limit_of_Tail_of_Decreasing_Sequence_of_Sets | [
"Limits of Sets",
"Decreasing Sequences of Sets"
] | [
"Definition:Set",
"Definition:Decreasing Sequence of Sets",
"Definition:Limit of Decreasing Sequence of Sets"
] | [
"Tail of Decreasing Sequence of Sets is Decreasing",
"Definition:Decreasing Sequence of Sets",
"Intersection is Decreasing",
"Definition:Decreasing Sequence of Sets",
"Definition:Set Intersection",
"Definition:Subset",
"Category:Limits of Sets",
"Category:Decreasing Sequences of Sets"
] |
proofwiki-19668 | Tail of Decreasing Sequence of Sets is Decreasing | Let $X$ be a set.
Let $\sequence {E_n}_{n \mathop \in \N}$ be a decreasing sequence of subsets of $X$.
Then for each $m \in \N$ we have:
:$\sequence {E_{n + m} }_{n \mathop \in \N}$ is a decreasing sequence of sets. | Since $\sequence {E_n}_{n \mathop \in \N}$ is an decreasing sequence of sets, we have:
:$E_{n + 1} \subseteq E_n$ for each $n \in \N$.
Swapping $n$ for $n + m$, this in particular gives:
:$E_{n + m + 1} \subseteq E_{n + m}$ for each $n \in \N$.
So $\sequence {E_{n + m} }_{n \mathop \in \N}$ is a decreasing sequence of... | Let $X$ be a [[Definition:Set|set]].
Let $\sequence {E_n}_{n \mathop \in \N}$ be a [[Definition:Decreasing Sequence of Sets|decreasing sequence of subsets]] of $X$.
Then for each $m \in \N$ we have:
:$\sequence {E_{n + m} }_{n \mathop \in \N}$ is a [[Definition:Decreasing Sequence of Sets|decreasing sequence of set... | Since $\sequence {E_n}_{n \mathop \in \N}$ is an [[Definition:Decreasing Sequence of Sets|decreasing sequence of sets]], we have:
:$E_{n + 1} \subseteq E_n$ for each $n \in \N$.
Swapping $n$ for $n + m$, this in particular gives:
:$E_{n + m + 1} \subseteq E_{n + m}$ for each $n \in \N$.
So $\sequence {E_{n + m} }_... | Tail of Decreasing Sequence of Sets is Decreasing | https://proofwiki.org/wiki/Tail_of_Decreasing_Sequence_of_Sets_is_Decreasing | https://proofwiki.org/wiki/Tail_of_Decreasing_Sequence_of_Sets_is_Decreasing | [
"Decreasing Sequences of Sets"
] | [
"Definition:Set",
"Definition:Decreasing Sequence of Sets",
"Definition:Decreasing Sequence of Sets"
] | [
"Definition:Decreasing Sequence of Sets",
"Definition:Decreasing Sequence of Sets",
"Category:Decreasing Sequences of Sets"
] |
proofwiki-19669 | Measure of Limit of Decreasing Sequence of Measurable Sets/Corollary | Let $F \in \Sigma$.
Let $\sequence {F_n}_{n \mathop \in \N}$ be an decreasing sequence of $\Sigma$-measurable sets such that:
:$F_n \downarrow F$
where $F_n \downarrow F$ denotes the limit of decreasing sequence of sets.
Suppose also that $\map \mu {F_m} < \infty$ for some $m \in \N$.
Then:
:$\ds \map \mu F = \lim_... | Define the sequence $\sequence {E_n}_{n \mathop \in \N}$ by:
:$E_n = F_{m + n}$
Then from Tail of Decreasing Sequence of Sets is Decreasing:
:$\sequence {E_n}_{\mathop \in \N}$ is an decreasing sequence of $\Sigma$-measurable sets.
From Limit of Tail of Decreasing Sequence of Sets, we have:
:$E_n \downarrow F$
with:... | Let $F \in \Sigma$.
Let $\sequence {F_n}_{n \mathop \in \N}$ be an [[Definition:Decreasing Sequence of Sets|decreasing sequence]] of [[Definition:Measurable Set|$\Sigma$-measurable sets]] such that:
:$F_n \downarrow F$
where $F_n \downarrow F$ denotes the [[Definition:Limit of Decreasing Sequence of Sets|limit of ... | Define the [[Definition:Sequence|sequence]] $\sequence {E_n}_{n \mathop \in \N}$ by:
:$E_n = F_{m + n}$
Then from [[Tail of Decreasing Sequence of Sets is Decreasing]]:
:$\sequence {E_n}_{\mathop \in \N}$ is an [[Definition:Decreasing Sequence of Sets|decreasing sequence]] of [[Definition:Measurable Set|$\Sigma$-m... | Measure of Limit of Decreasing Sequence of Measurable Sets/Corollary | https://proofwiki.org/wiki/Measure_of_Limit_of_Decreasing_Sequence_of_Measurable_Sets/Corollary | https://proofwiki.org/wiki/Measure_of_Limit_of_Decreasing_Sequence_of_Measurable_Sets/Corollary | [
"Measure of Limit of Decreasing Sequence of Measurable Sets"
] | [
"Definition:Decreasing Sequence of Sets",
"Definition:Measurable Set",
"Definition:Limit of Decreasing Sequence of Sets"
] | [
"Definition:Sequence",
"Tail of Decreasing Sequence of Sets is Decreasing",
"Definition:Decreasing Sequence of Sets",
"Definition:Measurable Set",
"Limit of Tail of Decreasing Sequence of Sets",
"Definition:Decreasing Sequence of Sets",
"Measure is Monotone",
"Measure of Limit of Decreasing Sequence o... |
proofwiki-19670 | Distribution Function of Finite Borel Measure is Bounded | Let $\mu$ be a finite Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then $F_\mu$ is bounded. | Let $x \in \R$.
Then:
:$\O \subseteq \hointl {-\infty} x \subseteq \R$
Then, from Measure is Monotone we have:
:$\map \mu \O \le \map \mu {\hointl {-\infty} x} \le \map \mu \R$
From Empty Set is Null Set, we have:
:$\map \mu \O = 0$
Since $\mu$ is a finite measure, we have:
:$\map \mu \R < \infty$
So, we have:
:$0... | Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]] on $\R$.
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Borel Measure|distribution function]] of $\mu$.
Then $F_\mu$ is [[Definition:Bounded Real-Valued Function|bounded]]. | Let $x \in \R$.
Then:
:$\O \subseteq \hointl {-\infty} x \subseteq \R$
Then, from [[Measure is Monotone]] we have:
:$\map \mu \O \le \map \mu {\hointl {-\infty} x} \le \map \mu \R$
From [[Empty Set is Null Set]], we have:
:$\map \mu \O = 0$
Since $\mu$ is a [[Definition:Finite Measure|finite measure]], we hav... | Distribution Function of Finite Borel Measure is Bounded | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Bounded | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Bounded | [
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure",
"Definition:Bounded Mapping/Real-Valued"
] | [
"Measure is Monotone",
"Measure of Empty Set is Zero",
"Definition:Finite Measure"
] |
proofwiki-19671 | Distribution Function of Finite Borel Measure is Increasing | Let $\mu$ be a finite Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then $F_\mu$ is an increasing function. | Let $x, y \in \R$ be such that $x \le y$.
Then:
:$\hointl {-\infty} x \subseteq \hointl {-\infty} x$
So, from Measure is Monotone:
:$\map \mu {\hointl {-\infty} x} \le \map \mu {\hointl {-\infty} y}$
That is:
:$\map {F_\mu} x \le \map {F_\mu} y$ whenever $x \le y$.
So $F_\mu$ is an increasing function.
{{qed}} | Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]] on $\R$.
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Borel Measure|distribution function]] of $\mu$.
Then $F_\mu$ is an [[Definition:Increasing Function|increasing function]]. | Let $x, y \in \R$ be such that $x \le y$.
Then:
:$\hointl {-\infty} x \subseteq \hointl {-\infty} x$
So, from [[Measure is Monotone]]:
:$\map \mu {\hointl {-\infty} x} \le \map \mu {\hointl {-\infty} y}$
That is:
:$\map {F_\mu} x \le \map {F_\mu} y$ whenever $x \le y$.
So $F_\mu$ is an [[Definition:Increasing ... | Distribution Function of Finite Borel Measure is Increasing | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Increasing | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Increasing | [
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure",
"Definition:Increasing/Real Function"
] | [
"Measure is Monotone",
"Definition:Increasing/Real Function"
] |
proofwiki-19672 | Distribution Function of Finite Borel Measure is Right-Continuous | Let $\mu$ be a finite Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then $F_\mu$ is right-continuous. | From Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals, it suffices to show that:
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, that converge to $x$ we have:
::$\ds \map {F_\mu} x = \lim_{n \mathop \to \infty} \map {F_\mu} {x_n}$
Now l... | Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]] on $\R$.
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Borel Measure|distribution function]] of $\mu$.
Then $F_\mu$ is [[Definition:Right-Continuous Real Function|right-continuous]]. | From [[Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals|Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals]], it suffices to show that:
:for all [[Definition:Monotone Real Sequence|monotone sequences]] $\sequence {x_n}_{n \mathop \in \N}$, with $x_... | Distribution Function of Finite Borel Measure is Right-Continuous | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Right-Continuous | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Right-Continuous | [
"Right-Continuous Functions",
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure",
"Definition:Continuous Real Function/Right-Continuous"
] | [
"Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Decreasing Sequence of Sets",
"Limit ... |
proofwiki-19673 | Limit of Distribution Function of Finite Borel Measure at Negative Infinity | Let $\mu$ be a finite Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map {F_\mu} x = 0$ | From Sequential Characterisation of Limit at Minus Infinity of Real Function: Corollary, we aim to show that:
:for all decreasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to -\infty$ we have $\ds \lim_{n \mathop \to \infty} \map {F_\mu} {x_n} = 0$.
Since $\sequence {x_n}_{n \mathop \in \N}$ is de... | Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]] on $\R$.
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Borel Measure|distribution function]] of $\mu$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map {F_\mu} x = 0$ | From [[Sequential Characterization of Limit at Minus Infinity of Real Function/Corollary|Sequential Characterisation of Limit at Minus Infinity of Real Function: Corollary]], we aim to show that:
:for all [[Definition:Decreasing Sequence|decreasing]] [[Definition:Real Sequence|real sequences]] $\sequence {x_n}_{n \ma... | Limit of Distribution Function of Finite Borel Measure at Negative Infinity | https://proofwiki.org/wiki/Limit_of_Distribution_Function_of_Finite_Borel_Measure_at_Negative_Infinity | https://proofwiki.org/wiki/Limit_of_Distribution_Function_of_Finite_Borel_Measure_at_Negative_Infinity | [
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure"
] | [
"Sequential Characterization of Limit at Minus Infinity of Real Function/Corollary",
"Definition:Decreasing/Sequence",
"Definition:Real Sequence",
"Definition:Decreasing/Sequence",
"Definition:Sequence",
"Definition:Decreasing Sequence of Sets",
"Limit of Decreasing Sequence of Unbounded Below Closed In... |
proofwiki-19674 | Properties of Limit at Infinity of Real Function/Sum Rule | Let $a \in \R$.
Let $f, g : \hointr a \infty \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then:
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x +... | Since:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
given $\epsilon > 0$, we can find $M_1 \ge 0$ such that:
:$\ds \size {\map f x - L_1} < \frac \epsilon 2$ for $x \ge M_1$
Since:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
we can find $M_2 \ge 0$ such that:
:$\ds \size {\map f x - L_2} < \frac \epsilon... | Let $a \in \R$.
Let $f, g : \hointr a \infty \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the [[Definition:Limit at Infinity|limit at $+\in... | Since:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
given $\epsilon > 0$, we can find $M_1 \ge 0$ such that:
:$\ds \size {\map f x - L_1} < \frac \epsilon 2$ for $x \ge M_1$
Since:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
we can find $M_2 \ge 0$ such that:
:$\ds \size {\map f x - L_2} < \frac \... | Properties of Limit at Infinity of Real Function/Sum Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Sum_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Sum_Rule | [
"Properties of Limit at Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Triangle Inequality/Real Numbers",
"Category:Properties of Limit at Infinity of Real Function"
] |
proofwiki-19675 | Properties of Limit at Infinity of Real Function/Multiple Rule | Let $a, \alpha \in \R$.
Let $f : \hointr a \infty \to \R$ be a real function such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop \to \inf... | If $\alpha = 0$, the result follows from Limit of Constant Function: Limit at $\infty$.
Otherwise, since:
:$\ds \lim_{x \mathop \to \infty} \map f x = L$
given $\epsilon > 0$ we can find a real number $M \ge 0$ such that:
:$\ds \size {\map f x - L} < \frac \epsilon {\size \alpha}$ for $x \ge M$.
Then:
:$\ds \size {\... | Let $a, \alpha \in \R$.
Let $f : \hointr a \infty \to \R$ be a [[Definition:Real Function|real function]] such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L$
where $\ds \lim_{x \mathop \to \infty}$ denotes the [[Definition:Limit at Infinity|limit at $+\infty$]].
Then:
:$\ds \lim_{x \mathop \to \infty} \... | If $\alpha = 0$, the result follows from [[Limit of Constant Function/Limit at Infinity|Limit of Constant Function: Limit at $\infty$]].
Otherwise, since:
:$\ds \lim_{x \mathop \to \infty} \map f x = L$
given $\epsilon > 0$ we can find a [[Definition:Real Number|real number]] $M \ge 0$ such that:
:$\ds \size {\ma... | Properties of Limit at Infinity of Real Function/Multiple Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Multiple_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Multiple_Rule | [
"Properties of Limit at Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Limit of Constant Function/Limit at Infinity",
"Definition:Real Number",
"Category:Properties of Limit at Infinity of Real Function"
] |
proofwiki-19676 | Properties of Limit at Infinity of Real Function/Combined Sum Rule | Let $a, \alpha, \beta \in \R$.
Let $f, g : \hointr a \infty \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then:
:$\ds \lim_{x \mathop \to \infty} \pa... | From Properties of Limit at Positive Infinity of Real Function: Multiple Rule, we have:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x} = \alpha L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \paren {\beta \map f x} = \beta L_2$
From Properties of Limit at Positive Infinity of Real Function: Sum Rule, we have:... | Let $a, \alpha, \beta \in \R$.
Let $f, g : \hointr a \infty \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the [[Definition:Limit at Infinity... | From [[Properties of Limit at Infinity of Real Function/Multiple Rule|Properties of Limit at Positive Infinity of Real Function: Multiple Rule]], we have:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x} = \alpha L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \paren {\beta \map f x} = \beta L_2$
From [[Prop... | Properties of Limit at Infinity of Real Function/Combined Sum Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Combined_Sum_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Combined_Sum_Rule | [
"Properties of Limit at Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Properties of Limit at Infinity of Real Function/Multiple Rule",
"Properties of Limit at Infinity of Real Function/Sum Rule",
"Category:Properties of Limit at Infinity of Real Function"
] |
proofwiki-19677 | Properties of Limit at Infinity of Real Function/Difference Rule | Let $a \in \R$.
Let $f, g : \hointr a \infty \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then:
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x -... | This follows immediately by plugging $\alpha = 1$ and $\beta = -1$ into the combined sum rule.
{{qed}}
Category:Properties of Limit at Infinity of Real Function
91ovt09v20uyg27b6fh16j2nb1y0dz7 | Let $a \in \R$.
Let $f, g : \hointr a \infty \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the [[Definition:Limit at Infinity|limit at $+\in... | This follows immediately by plugging $\alpha = 1$ and $\beta = -1$ into the [[Properties of Limit at Infinity of Real Function/Combined Sum Rule|combined sum rule]].
{{qed}}
[[Category:Properties of Limit at Infinity of Real Function]]
91ovt09v20uyg27b6fh16j2nb1y0dz7 | Properties of Limit at Infinity of Real Function/Difference Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Difference_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Difference_Rule | [
"Properties of Limit at Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Properties of Limit at Infinity of Real Function/Combined Sum Rule",
"Category:Properties of Limit at Infinity of Real Function"
] |
proofwiki-19678 | Properties of Limit at Infinity of Real Function/Product Rule | Let $a \in \R$.
Let $f, g : \hointr a \infty \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then:
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x \... | Write:
{{begin-eqn}}
{{eqn | l = \size {\map f x \map g x - L_1 L_2}
| r = \size {\map f x \map g x - L_1 L_2 + L_2 \map f x - L_2 \map f x}
}}
{{eqn | r = \size {\map f x \paren {\map g x - L_2} + L_2 \paren {\map f x - L_1} }
}}
{{eqn | o = \le
| r = \size {\map f x} \size {\map g x - L_2} + \size {L_2} \size {\... | Let $a \in \R$.
Let $f, g : \hointr a \infty \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the [[Definition:Limit at Infinity|limit at $+\in... | Write:
{{begin-eqn}}
{{eqn | l = \size {\map f x \map g x - L_1 L_2}
| r = \size {\map f x \map g x - L_1 L_2 + L_2 \map f x - L_2 \map f x}
}}
{{eqn | r = \size {\map f x \paren {\map g x - L_2} + L_2 \paren {\map f x - L_1} }
}}
{{eqn | o = \le
| r = \size {\map f x} \size {\map g x - L_2} + \size {L_2} \size {... | Properties of Limit at Infinity of Real Function/Product Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Product_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Infinity_of_Real_Function/Product_Rule | [
"Properties of Limit at Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Definition:Real Number",
"Reverse Triangle Inequality",
"Definition:Real Number",
"Definition:Real Number",
"Category:Properties of Limit at Infinity of Real Function"
] |
proofwiki-19679 | Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity | Let $a \in \R$.
Let $f : \hointl {-\infty} a \to \R$ be a real function.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ exists {{iff}} $\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists
and in this case:
:$\ds \lim_{x \mathop \to -\infty} \map f x = \lim_{x \mathop \to \infty} \map f {-x}$
where:
:$\ds \lim_{x \... | Note that:
:$\ds \lim_{x \mathop \to -\infty} \map f x = L$
{{iff}} given $\epsilon > 0$ we can find real number $M \le 0$ such that:
:$\size {\map f x - L} < \epsilon$ for $x \le M$.
This is equivalent to:
:given $\epsilon > 0$ we can find real number $M \le 0$ such that $\size {\map f x - L} < \epsilon$ for $x \g... | Let $a \in \R$.
Let $f : \hointl {-\infty} a \to \R$ be a [[Definition:Real Function|real function]].
Then:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ exists {{iff}} $\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists
and in this case:
:$\ds \lim_{x \mathop \to -\infty} \map f x = \lim_{x \mathop \to \infty} ... | Note that:
:$\ds \lim_{x \mathop \to -\infty} \map f x = L$
{{iff}} given $\epsilon > 0$ we can find [[Definition:Real Number|real number]] $M \le 0$ such that:
:$\size {\map f x - L} < \epsilon$ for $x \le M$.
This is equivalent to:
:given $\epsilon > 0$ we can find [[Definition:Real Number|real number]] $M \... | Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Relation_with_Limit_at_Infinity | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Relation_with_Limit_at_Infinity | [
"Properties of Limit at Minus Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Positive",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Category:Properties of Limit at Minus Infinity of Real Function"
] |
proofwiki-19680 | Cowen's Theorem | Let $g$ be a progressing mapping.
Let $x$ be a set.
Let $\powerset x$ denote the power set of $x$.
Let $M_x$ denote the intersection of the $x$-special subsets of $\powerset x$ {{WRT}} $g$.
Let $M$ be the class of all $x$ such that $x \in M_x$.
Then $M$ is minimally superinductive under $g$. | === Lemma $1$ ===
{{:Cowen's Theorem/Lemma 1}}{{qed|lemma}}
=== Lemma $2$ ===
{{:Cowen's Theorem/Lemma 2}}{{qed|lemma}}
=== Lemma $3$ ===
{{:Cowen's Theorem/Lemma 3}}{{qed|lemma}}
=== Lemma $4$ ===
{{:Cowen's Theorem/Lemma 4}}{{qed|lemma}}
=== Lemma $5$ ===
{{:Cowen's Theorem/Lemma 5}}{{qed|lemma}}
=== Lemma $6$ ===
{{... | Let $g$ be a [[Definition:Progressing Mapping|progressing mapping]].
Let $x$ be a [[Definition:Set|set]].
Let $\powerset x$ denote the [[Definition:Power Set|power set]] of $x$.
Let $M_x$ denote the [[Definition:Intersection of Special Subsets|intersection]] of the [[Definition:Special Set|$x$-special]] [[Definition... | === [[Cowen's Theorem/Lemma 1|Lemma $1$]] ===
{{:Cowen's Theorem/Lemma 1}}{{qed|lemma}}
=== [[Cowen's Theorem/Lemma 2|Lemma $2$]] ===
{{:Cowen's Theorem/Lemma 2}}{{qed|lemma}}
=== [[Cowen's Theorem/Lemma 3|Lemma $3$]] ===
{{:Cowen's Theorem/Lemma 3}}{{qed|lemma}}
=== [[Cowen's Theorem/Lemma 4|Lemma $4$]] ===
{{:C... | Cowen's Theorem/Proof | https://proofwiki.org/wiki/Cowen's_Theorem | https://proofwiki.org/wiki/Cowen's_Theorem/Proof | [
"Cowen's Theorem",
"Progressing Mappings",
"Superinductive Classes"
] | [
"Definition:Progressing Mapping",
"Definition:Set",
"Definition:Power Set",
"Definition:Intersection of Special Subsets",
"Definition:Special Set",
"Definition:Subset",
"Definition:Class (Class Theory)",
"Definition:Minimally Superinductive Class"
] | [
"Cowen's Theorem/Lemma 1",
"Cowen's Theorem/Lemma 2",
"Cowen's Theorem/Lemma 3",
"Cowen's Theorem/Lemma 4",
"Cowen's Theorem/Lemma 5",
"Cowen's Theorem/Lemma 6",
"Cowen's Theorem/Lemma 7",
"Cowen's Theorem/Lemma 2",
"Cowen's Theorem/Lemma 7",
"Definition:Closed Set under Progressing Mapping",
"C... |
proofwiki-19681 | Properties of Limit at Minus Infinity of Real Function/Sum Rule | Let $a \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x + ... | From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, we have:
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ and $\ds \lim_{x \mathop \to \infty} \map g {-x}$ exist.
From Properties of Limit at Infinity of Real Function: Sum Rule, we then have:
:$\ds \lim_{x \mathop \to \infty... | Let $a \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the [[Definition:Limit at Minus Infinity|limit at $-\... | From [[Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity|Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity]], we have:
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ and $\ds \lim_{x \mathop \to \infty} \map g {-x}$ exist.
From [[Properties o... | Properties of Limit at Minus Infinity of Real Function/Sum Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Sum_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Sum_Rule | [
"Properties of Limit at Minus Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity",
"Properties of Limit at Infinity of Real Function/Sum Rule",
"Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity",
"Category:Properties of Limit at Minus Infinity of Real Function"
] |
proofwiki-19682 | Properties of Limit at Minus Infinity of Real Function/Product Rule | Let $a \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x \m... | From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, we have:
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ and $\ds \lim_{x \mathop \to \infty} \map g {-x}$ exist.
From Properties of Limit at Infinity of Real Function: Product Rule, we then have:
:$\ds \lim_{x \mathop \to \i... | Let $a \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the [[Definition:Limit at Minus Infinity|limit at $-\... | From [[Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity|Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity]], we have:
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ and $\ds \lim_{x \mathop \to \infty} \map g {-x}$ exist.
From [[Properties o... | Properties of Limit at Minus Infinity of Real Function/Product Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Product_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Product_Rule | [
"Properties of Limit at Minus Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity",
"Properties of Limit at Infinity of Real Function/Product Rule",
"Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity",
"Category:Properties of Limit at Minus Infinity of Real Function"... |
proofwiki-19683 | Properties of Limit at Minus Infinity of Real Function/Multiple Rule | Let $a, \alpha \in \R$.
Let $f : \hointl {-\infty} a \to \R$ be a real function such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ exists
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop... | From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, we have:
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists.
From Properties of Limit at Infinity of Real Function: Multiple Rule, we then have:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f {-x} }$ exists
with:... | Let $a, \alpha \in \R$.
Let $f : \hointl {-\infty} a \to \R$ be a [[Definition:Real Function|real function]] such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ exists
where $\ds \lim_{x \mathop \to -\infty}$ denotes the [[Definition:Limit at Minus Infinity|limit at $-\infty$]].
Then:
:$\ds \lim_{x \mathop... | From [[Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity|Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity]], we have:
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists.
From [[Properties of Limit at Infinity of Real Function/Multiple Rul... | Properties of Limit at Minus Infinity of Real Function/Multiple Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Multiple_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Multiple_Rule | [
"Properties of Limit at Minus Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity",
"Properties of Limit at Infinity of Real Function/Multiple Rule",
"Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity",
"Category:Properties of Limit at Minus Infinity of Real Function... |
proofwiki-19684 | Properties of Limit at Minus Infinity of Real Function/Combined Sum Rule | Let $a, \alpha, \beta \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \par... | From Properties of Limit at Minus Infinity of Real Function: Multiple Rule:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x} = \alpha \lim_{x \mathop \to -\infty} \map f x$
and:
:$\ds \lim_{x \mathop \to -\infty} \paren {\beta \map g x... | Let $a, \alpha, \beta \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the [[Definition:Limit at Minus Infini... | From [[Properties of Limit at Minus Infinity of Real Function/Multiple Rule|Properties of Limit at Minus Infinity of Real Function: Multiple Rule]]:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x} = \alpha \lim_{x \mathop \to -\inf... | Properties of Limit at Minus Infinity of Real Function/Combined Sum Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Combined_Sum_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Combined_Sum_Rule | [
"Properties of Limit at Minus Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Properties of Limit at Minus Infinity of Real Function/Multiple Rule",
"Properties of Limit at Minus Infinity of Real Function/Sum Rule",
"Category:Properties of Limit at Minus Infinity of Real Function"
] |
proofwiki-19685 | Properties of Limit at Minus Infinity of Real Function/Difference Rule | Let $a, \alpha, \beta \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \par... | This is the combined sum rule with $\alpha = 1$, $\beta = -1$.
{{qed}}
Category:Properties of Limit at Minus Infinity of Real Function
sh5s1iky4cueqryz6iv0ppoahoz2pje | Let $a, \alpha, \beta \in \R$.
Let $f, g : \hointl {-\infty} a \to \R$ be [[Definition:Real Function|real functions]] such that:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the [[Definition:Limit at Minus Infini... | This is the [[Properties of Limit at Minus Infinity of Real Function/Combined Sum Rule|combined sum rule]] with $\alpha = 1$, $\beta = -1$.
{{qed}}
[[Category:Properties of Limit at Minus Infinity of Real Function]]
sh5s1iky4cueqryz6iv0ppoahoz2pje | Properties of Limit at Minus Infinity of Real Function/Difference Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Difference_Rule | https://proofwiki.org/wiki/Properties_of_Limit_at_Minus_Infinity_of_Real_Function/Difference_Rule | [
"Properties of Limit at Minus Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Properties of Limit at Minus Infinity of Real Function/Combined Sum Rule",
"Category:Properties of Limit at Minus Infinity of Real Function"
] |
proofwiki-19686 | Limit of Constant Function/Two-Sided Limit at Real Number | Let $a, b \in \R$.
Define $f : \R \to \R$ by:
:$\map f x = a$ for each $x \in \R$.
Then:
:$\ds \lim_{x \mathop \to b} \map f x = a$
where $\ds \lim_{x \mathop \to b}$ denotes the limit as $x \to b$. | We have:
:$\size {\map f x - a} = 0$ for all $x \in \R$.
So for any $\epsilon > 0$ and $\delta > 0$, we have:
:$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $\size {x - b} < \delta$.
So by the definition of the limit as $x \to b$, we have:
:$\ds \lim_{x \mathop \to b} \map f x = a$
{{qed}}
Category:Limi... | Let $a, b \in \R$.
Define $f : \R \to \R$ by:
:$\map f x = a$ for each $x \in \R$.
Then:
:$\ds \lim_{x \mathop \to b} \map f x = a$
where $\ds \lim_{x \mathop \to b}$ denotes the [[Definition:Limit of Real Function|limit as $x \to b$]]. | We have:
:$\size {\map f x - a} = 0$ for all $x \in \R$.
So for any $\epsilon > 0$ and $\delta > 0$, we have:
:$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $\size {x - b} < \delta$.
So by the definition of the [[Definition:Limit of Real Function|limit as $x \to b$]], we have:
:$\ds \lim_{x \mathop... | Limit of Constant Function/Two-Sided Limit at Real Number | https://proofwiki.org/wiki/Limit_of_Constant_Function/Two-Sided_Limit_at_Real_Number | https://proofwiki.org/wiki/Limit_of_Constant_Function/Two-Sided_Limit_at_Real_Number | [
"Limit of Constant Function"
] | [
"Definition:Limit of Real Function"
] | [
"Definition:Limit of Real Function",
"Category:Limit of Constant Function"
] |
proofwiki-19687 | Limit of Constant Function/Limit at Infinity | Let $a, b \in \R$.
Define $f : \R \to \R$ by:
:$\map f x = a$ for each $x \in \R$.
Then:
:$\ds \lim_{x \mathop \to \infty} \map f x = a$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$. | We have:
:$\size {\map f x - a} = 0$ for all $x \in \R$.
So for any $\epsilon > 0$ and $M \in \R$, we have:
:$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $x \ge M$.
So from the definition of the limit at $+\infty$, we have:
:$\ds \lim_{x \mathop \to \infty} \map f x = a$
{{qed}}
Category:Limit of Consta... | Let $a, b \in \R$.
Define $f : \R \to \R$ by:
:$\map f x = a$ for each $x \in \R$.
Then:
:$\ds \lim_{x \mathop \to \infty} \map f x = a$
where $\ds \lim_{x \mathop \to \infty}$ denotes the [[Definition:Limit at Infinity|limit at $+\infty$]]. | We have:
:$\size {\map f x - a} = 0$ for all $x \in \R$.
So for any $\epsilon > 0$ and $M \in \R$, we have:
:$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $x \ge M$.
So from the definition of the [[Definition:Limit at Infinity|limit at $+\infty$]], we have:
:$\ds \lim_{x \mathop \to \infty} \map f x... | Limit of Constant Function/Limit at Infinity | https://proofwiki.org/wiki/Limit_of_Constant_Function/Limit_at_Infinity | https://proofwiki.org/wiki/Limit_of_Constant_Function/Limit_at_Infinity | [
"Limit of Constant Function",
"Limit at Infinity"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive",
"Category:Limit of Constant Function",
"Category:Limit at Infinity"
] |
proofwiki-19688 | Limit of Constant Function/Limit at Minus Infinity | Let $a, b \in \R$.
Define $f : \R \to \R$ by:
:$\map f x = a$ for each $x \in \R$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map f x = a$
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$. | We have:
:$\size {\map f x - a} = 0$ for all $x \in \R$.
So for any $\epsilon > 0$ and $M \in \R$, we have:
:$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $x \le M$.
So from the definition of the limit at $-\infty$, we have:
:$\ds \lim_{x \mathop \to -\infty} \map f x = a$
{{qed}}
Category:Limit of Const... | Let $a, b \in \R$.
Define $f : \R \to \R$ by:
:$\map f x = a$ for each $x \in \R$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map f x = a$
where $\ds \lim_{x \mathop \to -\infty}$ denotes the [[Definition:Limit at Minus Infinity|limit at $-\infty$]]. | We have:
:$\size {\map f x - a} = 0$ for all $x \in \R$.
So for any $\epsilon > 0$ and $M \in \R$, we have:
:$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $x \le M$.
So from the definition of the [[Definition:Limit at Minus Infinity|limit at $-\infty$]], we have:
:$\ds \lim_{x \mathop \to -\infty} \... | Limit of Constant Function/Limit at Minus Infinity | https://proofwiki.org/wiki/Limit_of_Constant_Function/Limit_at_Minus_Infinity | https://proofwiki.org/wiki/Limit_of_Constant_Function/Limit_at_Minus_Infinity | [
"Limit at Minus Infinity",
"Limit of Constant Function",
"Limit at Minus Infinity"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Negative",
"Category:Limit of Constant Function",
"Category:Limit at Minus Infinity"
] |
proofwiki-19689 | Limit to Negative Infinity of Distribution Function of Finite Signed Borel Measure | Let $\mu$ be a finite signed Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map {F_\mu} x = 0$
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$. | Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
From Decomposition of Distribution Function of Finite Signed Borel Measure, we have:
:$F_\mu = F_{\mu^+} - F_{\mu^-}$
where $F_{\mu^+}$ and $F_{\mu^-}$ are the distribution functions of $\mu^+$ and $\mu^-$ respectively.
From Limit of Distribution Functi... | Let $\mu$ be a [[Definition:Finite Signed Measure|finite]] [[Definition:Signed Borel Measure|signed Borel measure]] on $\R$.
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Signed Borel Measure|distribution function]] of $\mu$.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map {F_\mu} x = 0$
where $\... | Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
From [[Decomposition of Distribution Function of Finite Signed Borel Measure]], we have:
:$F_\mu = F_{\mu^+} - F_{\mu^-}$
where $F_{\mu^+}$ and $F_{\mu^-}$ are the [[Definition:Distribution Function of Finite Borel... | Limit to Negative Infinity of Distribution Function of Finite Signed Borel Measure | https://proofwiki.org/wiki/Limit_to_Negative_Infinity_of_Distribution_Function_of_Finite_Signed_Borel_Measure | https://proofwiki.org/wiki/Limit_to_Negative_Infinity_of_Distribution_Function_of_Finite_Signed_Borel_Measure | [
"Distribution Function of Finite Signed Borel Measure"
] | [
"Definition:Finite Measure/Signed Measure",
"Definition:Signed Borel Measure",
"Definition:Distribution Function of Finite Signed Borel Measure",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Definition:Jordan Decomposition",
"Decomposition of Distribution Function of Finite Signed Borel Measure",
"Definition:Distribution Function of Finite Borel Measure",
"Limit of Distribution Function of Finite Borel Measure at Negative Infinity",
"Properties of Limit at Minus Infinity of Real Function/Diffe... |
proofwiki-19690 | Equivalence of Definitions of Primitive Root of Unity | Let $n \in \Z_{> 0}$ be a strictly positive integer.
Let $F$ be a field.
Let $U_n$ denote the set of all $n$-th roots of unity.
{{TFAE|def=Primitive Root of Unity}}
=== Definition 1 ===
{{:Definition:Primitive Root of Unity/Definition 1}}
=== Definition 2 ===
{{:Definition:Primitive Root of Unity/Definition 2}} | === Definition 1 implies Definition 2 ===
Let $\alpha \in U_n$ such that:
:$U_n = \set{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}}$
By {{Defof|Power of Field Element}}:
:$\alpha = \alpha^1$
Hence:
:$U_n = \set{1, \alpha^1, \alpha^2, \ldots, \alpha^{n-1}}$
By {{Defof|Explicit Set Definition}}:
:$\forall m \in \N: 0 < m <... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $U_n$ denote the [[Definition:Set|set]] of all [[Definition:Root of Unity|$n$-th roots of unity]].
{{TFAE|def=Primitive Root of Unity}}
=== [[Definition:Primi... | === Definition 1 implies Definition 2 ===
Let $\alpha \in U_n$ such that:
:$U_n = \set{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}}$
By {{Defof|Power of Field Element}}:
:$\alpha = \alpha^1$
Hence:
:$U_n = \set{1, \alpha^1, \alpha^2, \ldots, \alpha^{n-1}}$
By {{Defof|Explicit Set Definition}}:
:$\forall m \in \N: 0 < ... | Equivalence of Definitions of Primitive Root of Unity | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Primitive_Root_of_Unity | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Primitive_Root_of_Unity | [
"Primitive Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Field (Abstract Algebra)",
"Definition:Set",
"Definition:Root of Unity",
"Definition:Primitive Root of Unity/Definition 1",
"Definition:Primitive Root of Unity/Definition 2"
] | [] |
proofwiki-19691 | Measure of Half-Open Interval as Difference of Distribution Function | Let $a, b \in \R$.
Let $\mu$ be a finite Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then:
:$\map \mu {\hointl a b} = \map {F_\mu} b - \map {F_\mu} a$ | Note that:
:$\map \mu {\hointl {-\infty} a} < \infty$
since $\mu$ is finite.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {F_\mu} b - \map {F_\mu} a
| r = \map \mu {\hointl {-\infty} b} - \map \mu {\hointl {-\infty} a}
}}
{{eqn | r = \map \mu {\hointl {-\infty} b \setminus \hointl {-\infty} a}
| c = Measure of S... | Let $a, b \in \R$.
Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]] on $\R$.
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Borel Measure|distribution function]] of $\mu$.
Then:
:$\map \mu {\hointl a b} = \map {F_\mu} b - \map {F_\mu} a$ | Note that:
:$\map \mu {\hointl {-\infty} a} < \infty$
since $\mu$ is [[Definition:Finite Measure|finite]].
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {F_\mu} b - \map {F_\mu} a
| r = \map \mu {\hointl {-\infty} b} - \map \mu {\hointl {-\infty} a}
}}
{{eqn | r = \map \mu {\hointl {-\infty} b \setminus \hointl... | Measure of Half-Open Interval as Difference of Distribution Function | https://proofwiki.org/wiki/Measure_of_Half-Open_Interval_as_Difference_of_Distribution_Function | https://proofwiki.org/wiki/Measure_of_Half-Open_Interval_as_Difference_of_Distribution_Function | [
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Measure of Set Difference with Subset",
"Difference of Unbounded Closed Intervals"
] |
proofwiki-19692 | Difference of Unbounded Closed Intervals | Let $a, b \in \R$ have $a < b$.
Then:
:$\hointl {-\infty} b \setminus \hointl {-\infty} a = \hointl a b$
where $\setminus$ denotes set difference. | Note that:
:$x \in \hointl {-\infty} b \setminus \hointl {-\infty} a$
{{iff}}:
:$x \in \hointl {-\infty} b$ but $x \not \in \hointl {-\infty} a$.
That is:
:$x \le b$ but it is not the case that $x \le a$.
So this is equivalent to:
:$x \le b$ and $x > a$.
That is:
:$x \in \hointl a b$
So:
:$\hointl {-\infty} b \se... | Let $a, b \in \R$ have $a < b$.
Then:
:$\hointl {-\infty} b \setminus \hointl {-\infty} a = \hointl a b$
where $\setminus$ denotes [[Definition:Set Difference|set difference]]. | Note that:
:$x \in \hointl {-\infty} b \setminus \hointl {-\infty} a$
{{iff}}:
:$x \in \hointl {-\infty} b$ but $x \not \in \hointl {-\infty} a$.
That is:
:$x \le b$ but it is not the case that $x \le a$.
So this is equivalent to:
:$x \le b$ and $x > a$.
That is:
:$x \in \hointl a b$
So:
:$\hointl {-\i... | Difference of Unbounded Closed Intervals | https://proofwiki.org/wiki/Difference_of_Unbounded_Closed_Intervals | https://proofwiki.org/wiki/Difference_of_Unbounded_Closed_Intervals | [
"Real Intervals"
] | [
"Definition:Set Difference"
] | [
"Category:Real Intervals"
] |
proofwiki-19693 | Fundamental Theorem of Riemannian Geometry | Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.
Let $\nabla$ be a Levi-Civita connection.
Then $\nabla$ is unique. | {{ProofWanted}}
{{Namedfor|Bernhard Riemann|cat = Riemann}} | Let $\struct {M, g}$ be a [[Definition:Riemannian Manifold|Riemannian]] or [[Definition:Pseudo-Riemannian Manifold|pseudo-Riemannian manifold]] with or without [[Definition:Boundary (Topology)|boundary]].
Let $\nabla$ be a [[Definition:Levi-Civita Connection|Levi-Civita connection]].
Then $\nabla$ is [[Definition:Un... | {{ProofWanted}}
{{Namedfor|Bernhard Riemann|cat = Riemann}} | Fundamental Theorem of Riemannian Geometry | https://proofwiki.org/wiki/Fundamental_Theorem_of_Riemannian_Geometry | https://proofwiki.org/wiki/Fundamental_Theorem_of_Riemannian_Geometry | [
"Connections",
"Fundamental Theorems"
] | [
"Definition:Riemannian Manifold",
"Definition:Pseudo-Riemannian Manifold",
"Definition:Boundary (Topology)",
"Definition:Levi-Civita Connection",
"Definition:Unique"
] | [] |
proofwiki-19694 | Limit of Decreasing Sequence of Left Half-Open Intervals with Lower Bound Converging to Upper Bound | Let $a, b \in \R$ have $a < b$.
Let $\sequence {a_n}_{n \mathop \in \N}$ be an increasing sequence with $a_n \to b$.
Then we have:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b = \set b$
That is:
:$\hointl {a_n} b \downarrow \set b$
where $\downarrow$ denotes the limit of decreasing sequence of sets. | Clearly:
:$\ds b \in \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b$
so that:
:$\ds \set b \subseteq \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b$
Now let:
:$\ds x \in \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b$
Then:
:$a_n \le x \le b$
for each $n \in \N$.
From Limits Preserve Inequalities, we then have:
:$... | Let $a, b \in \R$ have $a < b$.
Let $\sequence {a_n}_{n \mathop \in \N}$ be an [[Definition:Increasing Sequence|increasing sequence]] with $a_n \to b$.
Then we have:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b = \set b$
That is:
:$\hointl {a_n} b \downarrow \set b$
where $\downarrow$ denotes the [[D... | Clearly:
:$\ds b \in \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b$
so that:
:$\ds \set b \subseteq \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b$
Now let:
:$\ds x \in \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b$
Then:
:$a_n \le x \le b$
for each $n \in \N$.
From [[Limits Preserve Inequalities]], we t... | Limit of Decreasing Sequence of Left Half-Open Intervals with Lower Bound Converging to Upper Bound | https://proofwiki.org/wiki/Limit_of_Decreasing_Sequence_of_Left_Half-Open_Intervals_with_Lower_Bound_Converging_to_Upper_Bound | https://proofwiki.org/wiki/Limit_of_Decreasing_Sequence_of_Left_Half-Open_Intervals_with_Lower_Bound_Converging_to_Upper_Bound | [
"Decreasing Sequences of Sets"
] | [
"Definition:Increasing/Sequence",
"Definition:Limit of Decreasing Sequence of Sets"
] | [
"Inequality Rule for Real Sequences",
"Category:Decreasing Sequences of Sets"
] |
proofwiki-19695 | Convolution Operator is Continuous Linear Transformation | Let $\map {L^1} \R$ and $\map {L^\infty} \R$ be the real Lebesgue $1$- and real Lebesgue $\infty$-spaces respectively.
Let $f \in \map {L^1} \R$.
Let $f* : \map {L^\infty} \R \to \map {L^\infty} \R$ be the convolution integral such that:
:$\ds \forall t \in \R : \forall g \in \map {L^\infty} \R : \map {\paren {f * g} ... | {{begin-eqn}}
{{eqn | ll= \forall t \in \R : \forall g \in \map {L^\infty} \R :
| l = \size {\map {\paren {f*g} } t}
| r = \size {\int_{-\infty}^\infty \map f {t - \tau} \map g \tau \rd \tau }
| c = {{Defof|Convolution Integral}}
}}
{{eqn | o = \le
| r = \int_{-\infty}^\infty \size {\map f {t - ... | Let $\map {L^1} \R$ and $\map {L^\infty} \R$ be the [[Definition:Lebesgue Space|real Lebesgue $1$-]] and [[Definition:L-Infinity|real Lebesgue $\infty$-spaces]] respectively.
Let $f \in \map {L^1} \R$.
Let $f* : \map {L^\infty} \R \to \map {L^\infty} \R$ be the [[Definition:Convolution Integral|convolution integral]... | {{begin-eqn}}
{{eqn | ll= \forall t \in \R : \forall g \in \map {L^\infty} \R :
| l = \size {\map {\paren {f*g} } t}
| r = \size {\int_{-\infty}^\infty \map f {t - \tau} \map g \tau \rd \tau }
| c = {{Defof|Convolution Integral}}
}}
{{eqn | o = \le
| r = \int_{-\infty}^\infty \size {\map f {t - ... | Convolution Operator is Continuous Linear Transformation | https://proofwiki.org/wiki/Convolution_Operator_is_Continuous_Linear_Transformation | https://proofwiki.org/wiki/Convolution_Operator_is_Continuous_Linear_Transformation | [
"Operator Theory",
"Improper Integrals",
"Linear Transformations",
"Continuous Transformations"
] | [
"Definition:Lebesgue Space",
"Definition:Lebesgue Space/L-Infinity",
"Definition:Convolution Integral",
"Definition:Continuous Linear Transformation Space"
] | [
"Integration by Substitution/Definite Integral",
"Definition:Element",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Bounded Mapping",
"Definition:Integral Operator",
"Integral Operator is Linear",
"Definition:Linear Operator",
"Continuity of Linear Transformation/Normed Vector Space",
... |
proofwiki-19696 | Tests for Finite Set Equality | Let $S$ and $T$ be finite sets.
Let $S \subseteq T$.
{{TFAE}}
:$(a): \quad \card S = \card T$
:$(b): \quad \card {T \setminus S} = 0$
:$(c): \quad T \setminus S = \O$
:$(d): \quad T = S$ | We have:
{{begin-eqn}}
{{eqn | n = a
| l = \card S
| r = \card T
}}
{{eqn | n = b
| ll = \leadstoandfrom
| l = \card {T \setminus S}
| r = \card T - \card S = 0
| c = Cardinality of Set Difference with Subset
}}
{{eqn | n = c
| ll = \leadstoandfrom
| l = T \setminus S
... | Let $S$ and $T$ be [[Definition:Finite Set|finite sets]].
Let $S \subseteq T$.
{{TFAE}}
:$(a): \quad \card S = \card T$
:$(b): \quad \card {T \setminus S} = 0$
:$(c): \quad T \setminus S = \O$
:$(d): \quad T = S$ | We have:
{{begin-eqn}}
{{eqn | n = a
| l = \card S
| r = \card T
}}
{{eqn | n = b
| ll = \leadstoandfrom
| l = \card {T \setminus S}
| r = \card T - \card S = 0
| c = [[Cardinality of Set Difference with Subset]]
}}
{{eqn | n = c
| ll = \leadstoandfrom
| l = T \setminus ... | Tests for Finite Set Equality | https://proofwiki.org/wiki/Tests_for_Finite_Set_Equality | https://proofwiki.org/wiki/Tests_for_Finite_Set_Equality | [
"Set Equality",
"Finite Sets"
] | [
"Definition:Finite Set"
] | [
"Cardinality of Set Difference with Subset",
"Cardinality of Empty Set",
"Set Difference with Superset is Empty Set",
"Category:Set Equality",
"Category:Finite Sets"
] |
proofwiki-19697 | Powers of Field Elements Commute | Let $\struct {F, +, \circ}$ be a field with zero $0_F$ and unity $1_F$.
Let $F^* = F \setminus {0_F}$ denote the set of elements of $F$ without the zero $0_F$.
Then:
:$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : a^m \circ a^n = a^n \circ a^m$
:$(b):\quad \forall a \in F : \forall n, m \in \Z_{\ge 0} : a^m \cir... | === Statement $(a)$ ===
By {{Defof|Field (Abstract Algebra)|Field}}:
:$\struct{F^*, \circ}$ is an Abelian group
By {{Defof|Power of Field Element}}:
:For all $a \in F^*$ and $n \in \Z$, $a^n$ is defined as the $n$th power of $a$ with respect to the Abelian group $\struct {F^*, \circ}$
From Powers of Group Element Commu... | Let $\struct {F, +, \circ}$ be a [[Definition:Field (Abstract Algebra)|field]] with [[Definition:Ring Zero|zero]] $0_F$ and [[Definition:Unity of Ring|unity]] $1_F$.
Let $F^* = F \setminus {0_F}$ denote the [[Definition:Set|set]] of [[Definition:Element|elements]] of $F$ without the [[Definition:Ring Zero|zero]] $0_F$... | === Statement $(a)$ ===
By {{Defof|Field (Abstract Algebra)|Field}}:
:$\struct{F^*, \circ}$ is an [[Definition:Abelian Group|Abelian group]]
By {{Defof|Power of Field Element}}:
:For all $a \in F^*$ and $n \in \Z$, $a^n$ is defined as the [[Definition:Power of Group Element|$n$th power of $a$]] with respect to the [[D... | Powers of Field Elements Commute | https://proofwiki.org/wiki/Powers_of_Field_Elements_Commute | https://proofwiki.org/wiki/Powers_of_Field_Elements_Commute | [
"Field Theory",
"Powers (Abstract Algebra)",
"Commutativity"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Set",
"Definition:Element",
"Definition:Ring Zero"
] | [
"Definition:Abelian Group",
"Definition:Power of Element/Group",
"Definition:Abelian Group",
"Powers of Group Element Commute"
] |
proofwiki-19698 | Index Laws/Negative Index/Field | :$\forall a \in \F^* : \forall n \in \Z : \paren{a^n}^{-1} = a^{-n} = \paren{a^{-1}}^n$ | By {{Defof|Field (Abstract Algebra)|Field}}:
:$\struct{F^*, \circ}$ is an Abelian group
By {{Defof|Power of Field Element}}:
:For all $a \in F^*$ and $n \in \Z$, $a^n$ is defined as the $n$th power of $a$ with respect to the Abelian group $\struct {F^*, \circ}$
From Negative Powers of Group Elements:
:$\forall a \in \F... | :$\forall a \in \F^* : \forall n \in \Z : \paren{a^n}^{-1} = a^{-n} = \paren{a^{-1}}^n$ | By {{Defof|Field (Abstract Algebra)|Field}}:
:$\struct{F^*, \circ}$ is an [[Definition:Abelian Group|Abelian group]]
By {{Defof|Power of Field Element}}:
:For all $a \in F^*$ and $n \in \Z$, $a^n$ is defined as the [[Definition:Power of Group Element|$n$th power of $a$]] with respect to the [[Definition:Abelian Group|... | Index Laws/Negative Index/Field | https://proofwiki.org/wiki/Index_Laws/Negative_Index/Field | https://proofwiki.org/wiki/Index_Laws/Negative_Index/Field | [
"Index Laws"
] | [] | [
"Definition:Abelian Group",
"Definition:Power of Element/Group",
"Definition:Abelian Group",
"Powers of Group Elements/Negative Index",
"Category:Index Laws"
] |
proofwiki-19699 | Index Laws/Sum of Indices/Field | :$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : a^m \circ a^n = a^\paren{m + n}$
:$(b):\quad \forall a \in \F : \forall n, m \in \Z_{\ge 0} : a^m \circ a^n = a^\paren{m + n}$ | === Statement $(a)$ ===
By {{Defof|Field (Abstract Algebra)|Field}}:
:$\struct{F^*, \circ}$ is an Abelian group
By {{Defof|Power of Field Element}}:
:For all $a \in F^*$ and $n \in \Z$, $a^n$ is defined as the $n$th power of $a$ with respect to the Abelian group $\struct {F^*, \circ}$
From Sum of Powers of Group Elemen... | :$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : a^m \circ a^n = a^\paren{m + n}$
:$(b):\quad \forall a \in \F : \forall n, m \in \Z_{\ge 0} : a^m \circ a^n = a^\paren{m + n}$ | === Statement $(a)$ ===
By {{Defof|Field (Abstract Algebra)|Field}}:
:$\struct{F^*, \circ}$ is an [[Definition:Abelian Group|Abelian group]]
By {{Defof|Power of Field Element}}:
:For all $a \in F^*$ and $n \in \Z$, $a^n$ is defined as the [[Definition:Power of Group Element|$n$th power of $a$]] with respect to the [[D... | Index Laws/Sum of Indices/Field | https://proofwiki.org/wiki/Index_Laws/Sum_of_Indices/Field | https://proofwiki.org/wiki/Index_Laws/Sum_of_Indices/Field | [
"Index Laws"
] | [] | [
"Definition:Abelian Group",
"Definition:Power of Element/Group",
"Definition:Abelian Group",
"Powers of Group Elements/Sum of Indices"
] |
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