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Lemma 86.3.9 (Properties obtained from filters)\n\nLet \( M \) be a transitive model of ZFC. If \( G \) is a filter, then \( M\left\lbrack G\right\rbrack \) is transitive and satisfies Extensionality, Foundation, EmptySet, Infinity, Pairing, and Union. | Proof. Hence, we get Extensionality and Foundation for free. Then Infinity and EmptySet follows from \( M \subseteq M\left\lbrack G\right\rbrack \) .\n\nFor Pairing, suppose \( {\sigma }_{1}^{G},{\sigma }_{2}^{G} \in M\left\lbrack G\right\rbrack \) . Then\n\n\[ \sigma = \left\{ {\left\langle {{\sigma }_{1},{1}_{\mathbb... | No |
Lemma 86.5.2 (Consistency and Persistence)\n\nWe have\n\n(1) (Consistency) If \( p \Vdash \varphi \) and \( q \leq p \) then \( q \Vdash \varphi \) .\n\n(2) (Persistence) If \( \{ q \mid q \Vdash \varphi \} \) is dense below \( p \) then \( p \Vdash \varphi \) . | You can prove both of these by induction on formula complexity. | No |
The set \( \{ p \mid p \Vdash \varphi \) or \( p \Vdash \neg \varphi \} \) is dense. | Proof. We claim that whenever \( p \nVdash \varphi \) then for some \( \bar{p} \leq p \) we have \( \bar{p} \Vdash \neg \varphi \) ; this will establish the corollary.\n\nBy the contrapositive of the previous lemma, \( \{ q \mid q \Vdash \varphi \} \) is not dense below \( p \), meaning for some \( \bar{p} \leq p \), e... | Yes |
Theorem 86.6.1 (The generic extension satisfies ZFC)\n\nSuppose \( M \) is a transitive model of ZFC. Let \( \mathbb{P} \in M \) be a poset, and \( G \subseteq \mathbb{P} \) is an \( M \) -generic filter. Then\n\n\[ M\left\lbrack G\right\rbrack \vDash \text{ZFC.} \] | Proof. We'll just do Comprehension, as the other remaining axioms are similar.\n\nSuppose \( {\sigma }^{G},{\sigma }_{1}^{G},\ldots ,{\sigma }_{n}^{G} \in M\left\lbrack G\right\rbrack \) are a set and parameters, and \( \varphi \left( {x,{x}_{1},\ldots ,{x}_{n}}\right) \) is a formula in the language of set theory. We ... | Yes |
Lemma 87.1.3 ( \( G \) encodes distinct real numbers)\n\nFor \( \alpha \in {\omega }_{2} \) define\n\n\[ \n{G}_{\alpha } = \{ n \mid G\left( {\alpha, n}\right) = 0\} \in \mathcal{P}\left( \mathbb{N}\right) .\n\]\n\nThen \( {G}_{\alpha } \neq {G}_{\beta } \) for any \( \alpha \neq \beta \) . | Proof. We claim that the set\n\n\[ \nD = \{ q \mid \exists n \in \omega : q\left( {\alpha, n}\right) \neq q\left( {\beta, n}\right) \text{ are both defined }\} \n\]\n\nis dense.\n\nQuestion 87.1.4. Check this. (Use the fact that the domains are all finite.)\n\nSince \( G \) is an \( M \) -generic it hits this dense set... | No |
Example 87.2.2 (Example of an antichain)\n\nIn the infinite binary tree, the set \( A = \{ {00},{01},{10},{11}\} \) is a strong antichain (in fact maximal by inclusion). | This is stronger than the notion of \ | No |
Lemma 87.2.5 (Possible values argument)\n\nSuppose \( M \) is a transitive model of ZFC and \( \mathbb{P} \) is a partial order such that \( \mathbb{P} \) has the \( \kappa \) -chain condition in \( M \) . Let \( X, Y \in M \) and let \( f : X \rightarrow Y \) be some function in \( M\left\lbrack G\right\rbrack \), but... | Proof. The idea behind the proof is easy: any possible value of \( f \) gives us some condition in the poset \( \mathbb{P} \) which forces it. Since distinct values must have incompatible conditions, the \( \kappa \) -chain condition guarantees there are at most \( \kappa \) such values.\n\nHere are the details. Let \(... | Yes |
Lemma 87.4.2 ( \( \Delta \) -System lemma)\n\nSuppose \( C \) is an uncountable collection of finite sets. Then \( \exists \bar{C} \subseteq C \) such that \( \bar{C} \) is an uncountable \( \Delta \) -system. | Proof. There exists an integer \( n \) such that \( C \) has uncountably many guys of length \( n \) . So we can throw away all the other sets, and just assume that all sets in \( C \) have size \( n \) .\n\nWe now proceed by induction on \( n \) . The base case \( n = 1 \) is trivial, since we can just take \( R = \va... | Yes |
Proposition 1.1 The set \( {S}_{E} \) of the finite sequences of elements of a countable set \( E \) is countable. | Proof Let \( \{ 2,3,5,7,{11},\ldots ,{m}_{j}\ldots \} \) be the sequence of prime numbers. Every positive integer \( n \) has a unique factorization, of the type\n\n\[ n = {2}^{{\alpha }_{1}}{3}^{{\alpha }_{2}}\cdots {m}_{j}^{{\alpha }_{j}} \]\n\nwhere the sequence \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{j}}\right... | Yes |
Corollary 1.2 The set \( \mathbb{Q} \) of the rational numbers is countable. | Proof The rational numbers can be put in one-to-one correspondence with a subset of the ratios \( \frac{m}{n} \) for two integers \( m, n \) with \( n \neq 0 \) . | No |
Proposition 1.2 The union of a countable collection of countable sets is countable. | Proof Let \( \left\{ {E}_{j}\right\} \) be a countable collection of countable sets. Since each of the \( {E}_{j} \) is countable, their elements may be listed as\n\n\[ \n{E}_{1} = \left\{ \begin{array}{llllll} {a}_{11} & {a}_{12} & {a}_{13} & \ldots & {a}_{1n} & \ldots \end{array}\right\} \n\]\n\n\[ \n{E}_{2} = \left\... | Yes |
Proposition 1.3 (Cantor [22]) The interval \( \left\lbrack {0,1}\right\rbrack \) as a subset of \( \mathbb{R} \), is not countable. | Proof The proof uses Cantor’s diagonal process. If \( \left\lbrack {0,1}\right\rbrack \) were countable, its elements could be listed as the countable collection\n\n\[ \n{x}_{j} = 0.{a}_{1j}{a}_{2j}\cdots {a}_{mj}\cdots \;\text{ for }j \in \mathbb{N} \n\]\n\nwhere \( {a}_{mj} \) are integers from 0 to 9 . Now set \( {a... | Yes |
Proposition 3.1 (Schöder-Bernstein) Let \( X \) and \( Y \) be any two sets. If \( \operatorname{card}\left( X\right) \leq \) \( \operatorname{card}\left( Y\right) \) and \( \operatorname{card}\left( X\right) \geq \operatorname{card}\left( Y\right) \), then \( \operatorname{card}\left( X\right) = \operatorname{card}\le... | Proof Let \( f \) and be a one-to-one function from \( X \) into \( Y \) and let \( g \) be a one-to-one function from \( Y \) into \( X \) . Partition \( X \) into the disjoint union of three sets \( {X}_{o},{X}_{1},{X}_{2} \) by the following iterative procedure. If \( x \) is not in the range of \( \mathbf{g} \) we ... | Yes |
Proposition 3.2 \( \operatorname{card}\left( {\mathbb{N}}^{m}\right) = \operatorname{card}\left( \mathbb{N}\right) \) and \( \operatorname{card}\left( {2}^{\mathbb{N}}\right) = \operatorname{card}\left( \mathbb{R}\right) \) . Moreover for any \( X \) there holds \( \operatorname{card}\left( {2}^{X}\right) > \operatorna... | Proof The first statement follows form Proposition 1.1. A non-empty subset \( A \subset \mathbb{N} \) , consists of an increasing sequence, finite or infinite, of positive integers, say for example \( A = \left\{ {{m}_{1},\ldots ,{m}_{n},\ldots }\right\} \) . Label by zero the elements of \( \mathbb{N} - A \), and by 1... | Yes |
Proposition 4.1 \( \operatorname{card}\left( {{2}^{\mathbb{N}} \times {2}^{\mathbb{N}}}\right) = \operatorname{card}\left( {2}^{\mathbb{N}}\right) = \operatorname{card}\left( \mathbb{R}\right) \) . | Proof We exhibit a one-to-one correspondence between \( \left( {{2}^{\mathbb{N}} \times {2}^{\mathbb{N}}}\right) \) and \( {2}^{\mathbb{N}} \) . For any two subsets \( A \) and \( B \) of \( \mathbb{N} \), set\n\n\[ g\left( A\right) = \mathop{\bigcup }\limits_{{n \in A}}\{ {2n}\} ,\;h\left( B\right) = \mathop{\bigcup }... | Yes |
Corollary 4.1 \( \operatorname{card}\left( {\mathbb{R}}^{m}\right) = \operatorname{card}\left( \mathbb{R}\right) \), for all \( m \in \mathbb{N} \) . | Proof Let first \( m = 2 \) . Then\n\n\[ \operatorname{card}\left( {\mathbb{R}}^{2}\right) = \operatorname{card}\left( {\mathbb{R} \times \mathbb{R}}\right) = \operatorname{card}\left( {{2}^{\mathbb{N}} \times {2}^{\mathbb{N}}}\right) = \operatorname{card}\left( \mathbb{R}\right) . \]\n\nFor general \( m \in \mathbb{N}... | No |
Proposition 4.2 Let \( X, Y \) and \( Z \) be any triple of sets. Then\n\n\[ \operatorname{card}\left( {X}^{Y \times Z}\right) = \operatorname{card}\left\lbrack {\left( {X}^{Y}\right) }^{Z}\right\rbrack \] | Proof Let \( f \in {X}^{Y \times Z} \) so that \( f\left( {y, z}\right) \in X \) for all pairs \( \left( {y, z}\right) \in Y \times Z \) . For a fixed \( z \in Z \), set \( {h}_{z}\left( y\right) = f\left( {y, z}\right) \) . This gives an element of \( {X}^{Y} \) . Thus as \( z \) ranges over \( Z \) , the map \( {h}_{... | Yes |
Corollary 4.2 \( \operatorname{card}\left( {\mathbb{R}}^{\mathbb{N}}\right) = \operatorname{card}\left( \mathbb{R}\right) \) . | Proof Since \( \mathbb{R} \) is in one-to-one correspondence with \( \{ 0,1{\} }^{\mathbb{N}} \)\n\n\[ \operatorname{card}\left( {\mathbb{R}}^{\mathbb{N}}\right) = \operatorname{card}\left\lbrack {\left( \{ 0,1{\} }^{\mathbb{N}}\right) }^{\mathbb{N}}\right\rbrack = \operatorname{card}\left( {\{ 0,1{\} }^{\mathbb{N} \ti... | Yes |
Corollary 4.3 \( \operatorname{card}\left( {\mathbb{N}}^{\mathbb{N}}\right) = \operatorname{card}\left( \mathbb{R}\right) \) . | Proof An element of \( {\mathbb{N}}^{\mathbb{N}} \) is a sequence of elements of \( \mathbb{N} \) . In particular \( {\mathbb{N}}^{\mathbb{N}} \) contains those sequences containing only two fixed elements of \( \mathbb{N} \), say for example \( \{ 1,2\} \) . Therefore \( \{ 1,2{\} }^{\mathbb{N}} \subset {\mathbb{N}}^{... | Yes |
Corollary 5.1 Let \( X \) be a set. There exists a function \( f : {2}^{X} \rightarrow X \) such that \( f\left( E\right) \in E \), for every \( E \subset X \) . Equivalently, one may choose an element out of every subset of \( X \) . | Proof Let \( f : {2}^{X} \rightarrow X \) be a function as in Corollary 5.1, whose existence is guaranteed by the axiom of choice. Set\n\n\[ \n{x}_{1} = f\left( X\right) \;\text{ and }\;{x}_{n} = f\left( {X - \mathop{\bigcup }\limits_{{j = 1}}^{{n - 1}}{x}_{j}}\right) \;\text{ for }n \geq 2.\n\]\n\nThe sequence \( \lef... | No |
Lemma 6.1 Every element of \( \mathcal{F} \) is a well ordering on its domain. | Proof Let \( D \subset X \) be the domain of a linear ordering \( \prec \in \mathcal{F} \) . For a nonempty subset \( A \subset D \) let\n\n\[ E = \{ y \in D \mid y \prec x\text{ for all }x \in A\text{ and }y \neq x\} . \]\n\nThen the first element of \( A \) is the first element of \( D - E \) and the latter is \( f\l... | No |
Lemma 6.2 Let \( { \prec }_{1} \) and \( { \prec }_{2} \) be two elements in \( \mathcal{F} \) with domains \( {D}_{1} \) and \( {D}_{2} \) . Then, one of the two domains, say for example \( {D}_{1} \) is a segment for the other, say for example \( {D}_{2} \) with respect to the corresponding ordering \( { \prec }_{2} ... | Proof Let \( E \) denote the set of all \( x \) such that\n\n\[ \left\{ {y \in {D}_{1} \mid y{ \prec }_{1}x}\right\} = \left\{ {y \in {D}_{2} \mid y{ \prec }_{2}x}\right\} \]\n\nand such that \( { \prec }_{1} \) and \( { \prec }_{2} \) agree on these two sets. By construction \( E \) is a segment for both \( {D}_{1} \)... | Yes |
Proposition 1.2 Let \( \{ X;\mathcal{U}\} \) be a Hausdorff topological space. Then, the points \( x \in X \) are closed. | Proof Every point \( y \in \left( {X - x}\right) \) is contained in some open set contained in \( X - x \) . Since \( X - x \) is the union of all such open sets, it is open. Thus \( \{ x\} \) is closed. | Yes |
Proposition 4.1 Let \( \mathcal{B} \) be a collection of sets in \( X \) satisfying (i)-(ii). There exists a collection \( \mathcal{U} \) of subsets of \( X \), which generates a topology on \( X \), for which \( \mathcal{B} \) is a base. | Proof Let \( \mathcal{U} \) consist of the empty set \( \varnothing \) and the collection of all subsets \( \mathcal{O} \) of \( X \), such that for every \( x \in \mathcal{O} \) there exists an element \( B \in \mathcal{B} \) such that \( x \in B \subset \mathcal{O} \) . Such a collection is not empty since \( X \in \... | Yes |
Proposition 4.2 Every topological space satisfying the second axiom of countability is separable. | Proof Let \( \{ \mathcal{O}\} \) be a countable base for the topology of \( \{ X;\mathcal{U}\} \) . For each \( i \in \mathbb{N} \) select an element \( {x}_{i} \in {\mathcal{O}}_{i} \) . This generates a countable, dense subset of \( \{ X;\mathcal{U}\} \) . | Yes |
Proposition 5.1 (i) \( \{ X;\mathcal{U}\} \) is compact if and only if every collection \( \mathcal{G} \) of closed sets with the finite intersection property has nonempty intersection. | Part (i) follows from the previous remarks. | No |
Proposition 5.2 (i) The continuous image of a countably compact space is countably compact. | Proof Parts (i)-(ii) follows from the definitions and the proof of Proposition 5.1. | No |
Proposition 5.3 Let \( \{ X;\mathcal{U}\} \) satisfy the second axiom of countability. Then every open covering of \( X \) contains a countable sub-covering. | Proof Let \( \left\{ {B}_{n}\right\} \) a countable collection of open sets that forms a base for the topology of \( \{ X;\mathcal{U}\} \) and let \( \mathcal{F} \) be an open covering of \( X \) . To each \( {B}_{n} \) we associate one and only one open set \( {\mathcal{O}}_{n} \in \mathcal{F} \) that contains it. The... | Yes |
Proposition 6.1 The closed interval \( \left\lbrack {0,1}\right\rbrack \) is compact. | Proof Let \( \left\{ {I}_{\alpha }\right\} \) be a collection of open intervals covering \( \left\lbrack {0,1}\right\rbrack \) and set\n\n\[ \mathcal{E} = \bigcup \left\{ \begin{matrix} x \in \left\lbrack {0,1}\right\rbrack \text{ such that the closed interval }\left\lbrack {0, x}\right\rbrack \text{ is } \\ \text{ cov... | Yes |
Proposition 6.2 The closed interval \( \left\lbrack {0,1}\right\rbrack \) has the Bolzano-Weierstrass property. | Proof Let \( \left\{ {x}_{n}\right\} \) be an infinite sequence of elements of \( \left\lbrack {0,1}\right\rbrack \) without a cluster point in \( \left\lbrack {0,1}\right\rbrack \) . Then, each of the open intervals \( \left( {x - \varepsilon, x + \varepsilon }\right) \), for \( x \in \left\lbrack {0,1}\right\rbrack \... | No |
Corollary 6.2 A bounded, closed subset \( E \subset {\mathbb{R}}^{N} \) has the Bolzano-Weierstrass property. | Proof Let \( \left\{ {x}_{n}\right\} \) be a sequence of elements in \( E \) and represent each of the \( {x}_{n} \) in terms of its coordinates, that is, \( {x}_{n} = \left( {{x}_{1, n},\ldots ,{x}_{N, n}}\right) \) . Since \( \left\{ {x}_{n}\right\} \) is bounded, each of the sequences \( \left\{ {x}_{j, n}\right\} \... | Yes |
Proposition 6.3 (Borel-Riesz ([18,124])) Let \( E \) be a bounded, closed subset of \( {\mathbb{R}}^{N} \). Then, every open covering \( \mathcal{U} \) of \( E \) contains a finite sub-covering \( {\mathcal{U}}^{\prime } \). | Proof By Proposition 5.3, may assume the covering is countable, say \( \mathcal{U} = \left\{ {\mathcal{O}}_{n}\right\} \). We claim that \( E \subset \mathop{\bigcup }\limits_{{i = 1}}^{m}{\mathcal{O}}_{i} \) for some \( m \in \mathbb{N} \). Indeed if not, we may select for each positive integer \( n \), an element \( ... | Yes |
Proposition 6.4 (Heine-Borel) Every compact subset of \( {\mathbb{R}}^{N} \), endowed with the Euclidean topology, is closed and bounded. | Proof Let \( E \subset {\mathbb{R}}^{N} \) be compact. Since \( {\mathbb{R}}^{N} \), endowed with the Euclidean topology, is a Hausdorff space, \( E \) is closed by (iii) of Proposition 5.1. The collection of balls \( \left\{ {B}_{n}\right\} \) centered at the origin and radius \( n \in \mathbb{N} \) is an open coverin... | Yes |
Theorem 7.1 (Weierstrass-Baire) Let \( \{ X;\mathcal{U}\} \) be countably compact and let \( f : X \rightarrow \) \( \mathbb{R} \) be upper semi-continuous. Then \( f \) is bounded above in \( X \) and it achieves its maximum in \( X \) . | Proof The collection of sets \( \{ \left\lbrack {f < n}\right\rbrack \} \) is a countable open covering of \( X \), from which we extract a finite one, say, for example, \( \left\lbrack {f < {n}_{1}}\right\rbrack ,\ldots ,\left\lbrack {f < {n}_{N}}\right\rbrack \) . Then \( f \leq \max \left\{ {{n}_{1},\ldots ,{n}_{N}}... | Yes |
Theorem 7.2 (Dini) Let \( \{ X;\mathcal{U}\} \) be countably compact and let \( \left\{ {f}_{n}\right\} \) be a sequence of real-valued, upper semi-continuous functions such that \( {f}_{n + 1} \leq {f}_{n} \) for all \( n \in \mathbb{N} \), and converging pointwise in \( X \) to a lower semi-continuous function \( f \... | Proof By possibly replacing \( {f}_{n} \) with \( {f}_{n} - f \), we may assume that \( \left\{ {f}_{n}\right\} \) is a decreasing sequence of upper semi-continuous functions converging to zero pointwise in \( X \) . For every \( \varepsilon > 0 \), the collection of open sets \( \left\lbrack {{f}_{n} < \varepsilon }\r... | Yes |
Theorem 8.1 (Tychonov ([165])) Let \( \left\{ {{X}_{\alpha };{\mathcal{U}}_{\alpha }}\right\} \) be a family of compact spaces. Then \( \prod {X}_{\alpha } \) endowed with the product topology, is compact. | The proof is based on showing that every collection of closed sets with the finite intersection property, has nonempty intersection. | No |
Lemma 8.1 Let \( \{ X;\mathcal{U}\} \) be a topological space and let \( {\mathcal{G}}_{o} \) be a collection of subsets of \( X \) with the finite intersection property. There exists a maximal collection \( \mathcal{G} \) of subsets of \( X \) with the finite intersection property and containing \( {\mathcal{G}}_{o} \... | Proof The family of all collections of sets with the finite intersection property and containing \( {\mathcal{G}}_{o} \) is partially ordered by inclusion, so that by the Hausdorff principle, there is a maximal linearly ordered subfamily \( \mathcal{F} \) . We claim that \( \mathcal{G} \) is the union of all the collec... | Yes |
Proposition 10.1 Let \( \{ X;\mathcal{U}\} \) be a topological vector space. Then:\n\n(i) The topology \( \mathcal{U} \) is generated by a symmetric base \( {\mathcal{B}}_{\Theta } \) .\n\n(ii) If \( \mathcal{O} \) is an open neighborhood of the origin, then \( X = \mathop{\bigcup }\limits_{{\lambda \in \mathbb{R}}}\la... | Proof The continuity of the multiplication by scalars implies that if \( \mathcal{O} \) is open, also \( \lambda \mathcal{O} \) is open for all \( \lambda \in \mathbb{R} - \{ 0\} \) . If \( \Theta \in \mathcal{O} \), then \( \Theta \in \lambda \mathcal{O} \) for all \( \left| \lambda \right| \leq 1 \) . In particular i... | Yes |
Proposition 10.2 Let \( \{ X;\mathcal{U}\} \) and \( \{ Y;\mathcal{V}\} \) be topological vector spaces. A linear map \( T : X \rightarrow Y \) is continuous if and only if is continuous at the origin \( \Theta \) of \( X \) . | Proof Since \( T \) is linear, \( T\left( \Theta \right) = \theta \in Y \), where \( \theta \) is the origin of \( Y \) . Let \( O \in \mathcal{V} \) be an open set containing \( \theta \) . By assumption \( {T}^{-1}\left( O\right) \) is an open set containing \( \Theta \) . Let \( x \in X \) be fixed. An open set in \... | Yes |
Proposition 10.3 A linear, continuous map T from a topological vector space \( \{ X;\mathcal{U}\} \) into a topological vector space \( \{ Y;\mathcal{V}\} \), is bounded. | Proof Let \( E \subset X \) be bounded. For every neighborhood \( O \) of the origin \( \theta \) of \( Y \), open in the topology of \( \{ Y;\mathcal{V}\} \), the inverse image \( {T}^{-1}\left( O\right) \) is a neighborhood of the origin \( \Theta \), open in the topology of \( \{ X;\mathcal{U}\} \) . Since \( E \) i... | Yes |
Proposition 11.1 Let \( T : \{ X;\mathcal{U}\} \rightarrow \mathbb{R} \) be a not identically zero, linear functional on \( X \) . Then:\n\n(i) If \( T \) is bounded in a neighborhood of the origin, then \( T \) is continuous.\n\n(ii) If \( \ker \{ T\} \) is closed then \( T \) is bounded in a neighborhood of the origi... | Proof Let \( \mathcal{O} \) be an open neighborhood of the origin such that \( \left| {T\left( x\right) }\right| \leq k \) for all \( x \in \mathcal{O} \) . For every \( \varepsilon \in \left( {0, k}\right) \) the pre-image of the open interval \( \left( {-\varepsilon ,\varepsilon }\right) \), contains the open sets \(... | Yes |
Proposition 11.2 Let \( \\left\\{ {{T}_{1},\\ldots ,{T}_{n}}\\right\\} \) be a finite collection of bounded linear functionals on a Hausdorff, linear, topological vector space \( \\{ X;\\mathcal{U}\\} \), and set\n\n\[ K = \\mathop{\\bigcap }\\limits_{{j = 1}}^{n}\\ker \\left\\{ {T}_{j}\\right\\} \]\n\nIf \( T \) is a ... | Proof The map\n\n\[ X \\ni x \\rightarrow \\left( {{T}_{1}\\left( x\\right) ,\\ldots ,{T}_{n}\\left( x\\right) }\\right) \\in {\\mathbb{R}}^{n} \]\n\nis bounded and linear, and its image \( {\\mathbb{R}}_{o}^{n} \) is a closed subspace \( {\\mathbb{R}}^{n} \) . The map\n\n\[ {\\mathbb{R}}_{o}^{n} \\ni \\left( {{T}_{1}\... | Yes |
Proposition 12.2 Let \( \{ X;\mathcal{U}\} \) be a Hausdorff, locally compact topological vector space. Then \( X \) is of finite dimension. | Proof Let \( \mathcal{O} \) be a neighborhood of the origin, whose closure is compact. We may assume that \( \mathcal{O} \) is symmetric and \( \lambda \mathcal{O} \subset \mathcal{O} \) for all \( \left| \lambda \right| \leq 1 \) . There exist at most finitely many points \( {x}_{1},\ldots ,{x}_{n} \in \mathcal{O} \),... | Yes |
Proposition 13.1 Let \( A \) be a subset of \( X \) . The function \( x \rightarrow d\left( {A, x}\right) \) is continuous in \( \{ X;d\} \) . | Proof Let \( x, y \in X \) and \( z \in A \) . By the requirement (iv) of a metric\n\n\[ d\left( {z, x}\right) \leq d\left( {x, y}\right) + d\left( {z, y}\right) . \]\n\nTaking the infimum of both sides for \( z \in A \) gives\n\n\[ d\left( {A, x}\right) \leq d\left( {x, y}\right) + d\left( {A, y}\right) . \]\n\nInterc... | Yes |
Proposition 13.2 A metric space \( \{ X;d\} \) is separable if and only if it satisfies the second axiom of countability. \( {}^{3} \) | Proof Let \( \{ X;d\} \) be separable and let \( A \) be a countable, dense subset of \( \{ X;d\} \) . The collection of balls centered at points of \( A \) and with rational radius forms a countable base for the topology of \( \{ X;d\} \) . The converse follows from Proposition 4.2. | Yes |
Corollary 13.1 Every subset of a separable metric space is separable. | Proof Let \( \left\{ {x}_{n}\right\} \) be a countable dense subset. For a pair of positive integers \( \left( {m, n}\right) \) , consider the balls \( {B}_{1/m}\left( {x}_{n}\right) \) centered at \( {x}_{n} \) and radius \( 1/m \) . If \( Y \) is a subset of \( X \) , the ball \( {B}_{1/m}\left( {x}_{n}\right) \) mus... | Yes |
Proposition 14.1 If \( d \) on a vector space \( X \) is translation invariant then the sum \( + : \left( {X \times X}\right) \rightarrow X \) is continuous. | Proof It suffices to show that \( X \times X \ni \left( {x, y}\right) \rightarrow x + y \) is continuous at an arbitrary point \( \left( {{x}_{o},{y}_{o}}\right) \in X \times X \) . From the definition of product topology\n\n\[ d\left( {x + y,{x}_{o} + {y}_{o}}\right) = d\left( {x - {x}_{o},{y}_{o} - y}\right) \]\n\n\[... | Yes |
Proposition 14.2 Let \( \{ X;d\} \) and \( \{ Y;\eta \} \) be metric vector spaces. A bounded linear map \( T : X \rightarrow Y \) is continuous. | Proof For any ball \( {\mathcal{B}}_{r} \) in \( \{ Y;\eta \} \), of radius \( r \) centered at the origin of \( Y \), there exists a ball \( {B}_{\rho } \) in \( \{ X;d\} \), centered at origin of \( X \) such that \( {B}_{\rho } \subset {T}^{-1}\left( {\mathcal{B}}_{r}\right) \) . If not, for all \( \delta > 0 \) the... | Yes |
Theorem 16.2 (Banach-Steinhaus [15]) Let \( \{ X;d\} \) be a complete metric space and let \( \mathcal{F} \) be a family of continuous, real-valued functions defined in \( X \) . Assume that the functions \( f \in \mathcal{F} \) are pointwise equi-bounded, that is, for all \( x \in X \), there exists a positive number ... | Proof For \( n \in \mathbb{N} \), let \( {E}_{n, f} \) and \( {E}_{n} \) be subsets of \( X \) defined by\n\n\[ {E}_{n, f} = \{ x \in X \mid \left| {f\left( x\right) }\right| \leq n\} ,\;{E}_{n} = \mathop{\bigcap }\limits_{{f \in \mathcal{F}}}{E}_{n, f}. \]\n\nThe sets \( {E}_{n, f} \) are closed, since the functions \... | Yes |
Corollary 17.1 A countably compact metric space is separable. | Proof For positive integers \( m \) and \( n \), let \( {E}_{n, m} \) be the finite \( \varepsilon \) -net of \( X \) corresponding to \( \varepsilon = \frac{1}{n} \) . The union \( \cup {E}_{n, m} \) is a countable subset of \( X \) which is dense in \( X \) . | Yes |
Proposition 17.2 A totally bounded and complete metric space \( \{ X;d\} \) is sequentially compact. | Proof Let \( \left\{ {x}_{n}\right\} \) be a sequence of elements of \( X \) . The proof consists of selecting a Cauchy sequence \( \left\{ {x}_{{n}^{\prime }}\right\} \subset \left\{ {x}_{n}\right\} \) . Since \( \{ X;d\} \) is complete, such a Cauchy subsequence would then converge to some \( x \in X \) thereby estab... | Yes |
Proposition 1.1 (Cantor [21]) Every open subset \( E \) of \( \mathbb{R} \) is the union of a countable collection of pairwise disjoint, open intervals. | Proof For \( x \in E \), let \( {E}_{x} \) be the union of all open intervals containing \( x \) and contained in \( E \) . By construction \( {E}_{x} \) is an interval. If \( x \) and \( y \) are two distinct elements of \( E \), then either \( {E}_{x} = {E}_{y} \) or \( {E}_{x} \cap {E}_{y} = \varnothing \) . Indeed ... | Yes |
Proposition 1.2 An open set \( E \subset {\mathbb{R}}^{N} \) is the union of a countable collection of \( \frac{1}{2} \) - closed disjoint, dyadic cubes. | Proof Consider the \( \frac{1}{2} \) -closed dyadic cubes \( {Q}_{1,\mathbf{q}} \) . At most countably many of them are contained in \( E \) and we denote by \( {\mathcal{Q}}_{1} \) their union, i.e.,\n\n\[ \n{\mathcal{Q}}_{1} = \left\{ {\bigcup {Q}_{1,\mathbf{q}} \mid {Q}_{1,\mathbf{q}} \subset E}\right\} \n\]\n\nIf t... | Yes |
Proposition 2.1 Given any collection \( \mathcal{O} \) of subsets of \( X \), there exists a smallest \( \sigma \) -algebra \( {\mathcal{A}}_{o} \) that contains \( \mathcal{O} \) . | Proof Let \( \mathcal{F} \) be the collection of all the \( \sigma \) -algebras containing \( \mathcal{O} \) . Such a collection is nonvoid since it contains discrete \( \sigma \) -algebra. Set\n\n\[ \n{\mathcal{A}}_{o} = \bigcap \{ \mathcal{A} \mid \mathcal{A} \in \mathcal{F}\} \n\]\n\nAny two sets in \( {\mathcal{A}}... | Yes |
Proposition 3.1 Let \( \mu : \mathcal{A} \rightarrow {\mathbb{R}}^{ * } \) be a measure and let \( A, B \in \mathcal{A} \) . Then:\n\n(a). \( \mu \) is monotone, that is \( \mu \left( A\right) \leq \mu \left( B\right) \), whenever \( A \subset B \) . | Proof Write \( B = A \cup \left( {B - A}\right) \) . Since \( A \) and \( B - A \) are disjoint, by (iii)\n\n\[ \mu \left( B\right) = \mu \left( A\right) + \mu \left( {B - A}\right) . \]\n\nThis proves the monotonicity of \( \mu \), since \( \mu \left( {B - A}\right) \geq 0 \) . | Yes |
Proposition 4.1 The set function \( {\mu }_{e} \) defined by (4.1) is an outer measure. | Proof The requirements (i)-(iii) are a direct consequence of the definitions. Let \( \left\{ {E}_{n}\right\} \) be a countable collection of subsets of \( X \) . In proving (iv) may assume that \( {\mu }_{e}\left( {E}_{n}\right) < \infty \) for all \( n \) . Fix \( \varepsilon > 0 \) . For each \( n \in \mathbb{N} \) t... | Yes |
Proposition 5.1 \( {\mathcal{H}}_{\alpha } \) is an outer measure on \( {\mathbb{R}}^{N} \) . Moreover,\n\n\[ \n{\mathcal{H}}_{\alpha }\left( E\right) < \infty \text{ implies }{\mathcal{H}}_{\beta }\left( E\right) = 0\;\text{ for all }\beta > \alpha ; \n\]\n\n\[ \n{\mathcal{H}}_{\alpha }\left( E\right) > 0\;\text{ impl... | Proof The first statement is proved as in Proposition 4.1. Let \( \left\{ {E}_{n}\right\} \) be countable collection of elements of \( {\mathcal{E}}_{\varepsilon } \) such that \( E \subset \bigcup {E}_{n} \) . For \( \beta > \alpha \)\n\n\[ \n{\mathcal{H}}_{\beta ,\varepsilon }\left( E\right) \leq \sum {\left( \operat... | Yes |
Proposition 5.2 Let \( {\mu }_{e}\left( \cdot \right) \) denote the Lebesgue outer measure in \( {\mathbb{R}}^{N} \) defined in (4.4). There exists two positive constants \( {c}_{N} \leq {C}_{N} \) depending only upon \( N \) such that, for every set \( E \subset {\mathbb{R}}^{N} \)\n\n\[ \n{c}_{N}{\mathcal{H}}_{N}\lef... | Proof May assume that both \( {\mu }_{e}\left( E\right) \) and \( {\mathcal{H}}_{N}\left( E\right) \) are finite.\n\nHaving fixed \( \varepsilon > 0 \) there exists a countable collection of \( \frac{1}{2} \) -closed dyadic cubes \( \left\{ {Q}_{n}\right\} \) whose union covers \( E \) and\n\n\[ \n{\mu }_{e}\left( E\ri... | Yes |
Proposition 5.3 Let \( E \) and \( F \) be subsets of \( {\mathbb{R}}^{N} \) such that \( \operatorname{dist}\{ E;F\} = \delta \) for some \( \delta > 0 \) . Then \( {\mathcal{H}}_{\alpha }\left( {E \cup F}\right) = {\mathcal{H}}_{\alpha }\left( E\right) + {\mathcal{H}}_{\alpha }\left( F\right) \), for all \( \alpha > ... | Proof Since \( {\mathcal{H}}_{\alpha } \) is subadditive, it suffices to prove the statement with equality replaced by \( \geq \) . Also may assume that \( {\mathcal{H}}_{\alpha }\left( {E \cup F}\right) \) is finite.\n\nHaving fixed \( \varepsilon < \frac{1}{2}\delta \), there exists a collection of sets \( \left\{ {G... | Yes |
Proposition 6.2 The restriction of \( {\mu }_{e} \) to \( \mathcal{A} \) is a complete measure. | Proof By Proposition 6.1, \( \mathcal{A} \) is a \( \sigma \) -algebra and the restriction \( {\left. {\mu }_{e}\right| }_{\mathcal{A}} \) satisfies the requirements of a complete measure. In particular, the countable additivity follows from (v) with \( A \) replaced with the union of the \( {E}_{n} \) and the complete... | Yes |
Proposition 7.1 \( {\mathcal{A}}_{f} \) contains the Borel sets on \( \mathbb{R} \) . | Proof It suffices to verify that intervals of the type \( (\alpha ,\beta \rbrack \) belong to \( {\mathcal{A}}_{f} \), i.e., that for every subset \( E \subset \mathbb{R} \)\n\n\[ \n{\mu }_{f, e}\left( E\right) \geq {\mu }_{f, e}(E \cap \left( {\alpha ,\beta \rbrack }\right) + {\mu }_{f, e}(E - \left( {\alpha ,\beta \r... | No |
Lemma 7.1 Let \( {\lambda }_{f} \) be the Lebesgue-Stieltjes set function defined on open intervals of \( \mathbb{R} \), from which \( {\mu }_{f, e} \) is constructed. Then for any open interval \( I \) and any interval of the type \( (\alpha ,\beta \rbrack \) , \[ {\lambda }_{f}\left( I\right) \geq {\mu }_{f, e}(I \ca... | Proof Let \( I = \left( {a, b}\right) \) and assume that \( (\alpha ,\beta \rbrack \subset \left( {a, b}\right) \) . Denote by \( \varepsilon \) and \( \eta \) positive numbers. By the right continuity of \( f \) and the definition of \( {\mu }_{f, e} \) , \[ {\lambda }_{f}\left( I\right) = f\left( b\right) - f\left( a... | Yes |
Proposition 8.1 \( {\mathcal{A}}_{\alpha } \) contains the Borel sets in \( {\mathbb{R}}^{N} \) . | Proof It suffices to verify that closed sets \( E \subset {\mathbb{R}}^{N} \) belong to \( {\mathcal{A}}_{\alpha } \), i.e., that for every subset \( A \subset {\mathbb{R}}^{N} \) ,\n\n\[ \n{\mathcal{H}}_{\alpha }\left( A\right) \geq {\mathcal{H}}_{\alpha }\left( {A \cap E}\right) + {\mathcal{H}}_{\alpha }\left( {A - E... | Yes |
Lemma 8.1 \( \lim {\mathcal{H}}_{\alpha }\left\lbrack {A \cap \left( {{E}_{n} - E}\right) }\right\rbrack = 0 \) . | Proof For \( j = n, n + 1,\ldots \), set\n\n\[ \n{F}_{j} = \left\{ {x \in A\left| {\;\frac{1}{j + 1} < \operatorname{dist}\{ x;E\} \leq \frac{1}{j}}\right. }\right\} .\n\]\n\nThen \( A \cap \left( {{E}_{n} - E}\right) = \mathop{\bigcup }\limits_{{j = n}}^{\infty }{F}_{j} \), and\n\n\[ \n{\mathcal{H}}_{\alpha }\left( {A... | No |
Proposition 10.1 Let \( E \subset X \) be of finite outer measure. For every \( \varepsilon > 0 \) there exists a set \( {E}_{\sigma ,\varepsilon } \in {\mathcal{Q}}_{\sigma } \) such that\n\n\[ E \subset {E}_{\sigma ,\varepsilon }\;\text{ and }\;{\mu }_{e}\left( E\right) \geq \mu \left( {E}_{\sigma ,\varepsilon }\righ... | Proof By (4.2), for every \( \varepsilon > 0 \) there exists \( {E}_{\sigma ,\varepsilon } = \bigcup {Q}_{n} \in {\mathcal{Q}}_{\sigma } \), such that\n\n\[ {\mu }_{e}\left( E\right) + \varepsilon \geq \sum \lambda \left( {Q}_{n}\right) = \sum \mu \left( {Q}_{n}\right) \geq \mu \left( {\bigcup {Q}_{n}}\right) = \mu \le... | Yes |
Proposition 10.2 Let \( \{ X,\mathcal{A},\mu \} \) be the measure space generated by a measure \( \lambda \) on a semi-algebra \( \mathcal{Q} \). A set \( E \subset X \) of finite outer measure, is \( \mu \)-measurable if and only if for every \( \varepsilon > 0 \) there exists a set \( {E}_{\sigma ,\varepsilon } \in {... | Proof The necessary condition follows from Proposition 10.1. Indeed if \( E \) is \( \mu \)- measurable \( \mu \left( {E}_{\sigma ,\varepsilon }\right) \leq {\mu }_{e}\left( E\right) + \varepsilon \; \Leftrightarrow \;\mu \left( {{E}_{\sigma ,\varepsilon } - E}\right) \leq \varepsilon . For the sufficient condition, as... | Yes |
Proposition 12.1 \( {\mu }_{e}\left( E\right) = {\mu }_{e}^{\prime }\left( E\right) \) . | Proof One may assume that \( {\mu }_{e}\left( E\right) < \infty \) . Having fixed \( \varepsilon > 0 \), let \( \left\{ {Q}_{\varepsilon, n}\right\} \) be a countable collection of \( \frac{1}{2} \) -closed, dyadic cubes, whose union contains \( E \) and satisfying (4.2). For each \( n \) there exists an open cube \( {... | Yes |
Proposition 12.2 A set \( E \subset {\mathbb{R}}^{N} \) such that \( {\mu }_{e}\left( E\right) < \infty \), is Lebesgue measurable if and only if for every \( \varepsilon > 0 \) there exists an open set \( {E}_{o,\varepsilon } \) such that\n\n\[ E \subset {E}_{o,\varepsilon }\;\text{ and }\;{\mu }_{e}\left( {{E}_{o,\va... | Proof The sufficient condition follows from Propositions 10.2 and Remark 10.1. Since the Lebesgue measure in \( {\mathbb{R}}^{N} \) is \( \sigma \) -finite, the necessary condition follows from Propositions 10.1-10.3, using that \( {\mu }_{e}\left( E\right) = {\mu }_{e}^{\prime }\left( E\right) \) . | Yes |
Proposition 12.3 A set \( E \subset {\mathbb{R}}^{N} \) such that \( {\mu }_{e}\left( E\right) < \infty \), is Lebesgue measurable if and only if for every \( \varepsilon > 0 \) there exists a closed set \( {E}_{c,\varepsilon } \) such that\n\n\[ \n{E}_{c,\varepsilon } \subset E\;\text{ and }\;{\mu }_{e}\left( {E - {E}... | Proof (Sufficient Condition) Assume that for all \( \varepsilon > 0 \) there exists a closed set \( {E}_{c,\varepsilon } \) satisfying (12.3). Then \( {E}_{c,\varepsilon }^{c} \) is open, it contains \( {E}^{c} \) and\n\n\[ \n{\mu }_{e}\left( {{E}_{c,\varepsilon }^{c} - {E}^{c}}\right) = {\mu }_{e}\left( {E - {E}_{c,\v... | Yes |
Proposition 15.1 Let \( \mu \) be a finite Borel measure in \( {\mathbb{R}}^{N} \) and let \( E \) be a Borel set. For every \( \varepsilon > 0 \) there exists a closed set \( {E}_{c,\varepsilon } \subset E \), such that\n\n\[ \n{E}_{c,\varepsilon } \subset E\;\text{ and }\;\mu \left( {E - {E}_{c,\varepsilon }}\right) ... | Proof (of (15.1)) Let \( \mathcal{A} \) be the \( \sigma \) -algebra where \( \mu \) is defined and set\n\n\[ \n{\mathcal{C}}_{o} = \left\{ \begin{matrix} \text{ the collection of sets }E \in \mathcal{A}\text{ such that for every }\varepsilon > 0 \\ \text{ there exists a closed set }C \subset E,\text{ such that }\mu \l... | Yes |
Proposition 16.1 Let \( \mu \) be a Borel measure in \( {\mathbb{R}}^{N} \) and let \( E \) be a Borel set of finite measure. For every \( \varepsilon > 0 \) there exists a closed set \( {E}_{c,\varepsilon } \subset E \), such that (15.1) holds. | Proof Let \( \mathcal{A} \) be the \( \sigma \) -algebra where \( \mu \) is defined. The set \( E \) being fixed, set \( {\mu }_{E}\left( A\right) = \) \( \mu \left( {E \cap A}\right) \) for all \( A \in \mathcal{A} \) . Then \( {\mu }_{E} \) is a finite Borel measure in \( {\mathbb{R}}^{N} \) . | No |
Proposition 16.2 Let \( \mu \) be a Borel measure in \( {\mathbb{R}}^{N} \) which is finite on bounded sets, and let \( E \) be a Borel set of finite measure. For every \( \varepsilon > 0 \) there exists an open set \( {E}_{o,\varepsilon } \supset E \), such that (15.2) holds. | Proof Let \( {Q}_{n} \) be the open cube centered at the origin, of edge \( n \) and with faces parallel to the coordinate planes. Since \( E \) is a Borel set, \( {Q}_{n} - E \) is also a Borel set and \( \mu \left( {{Q}_{n} - E}\right) < \infty \) . By Proposition 16.1, having fixed \( \varepsilon > 0 \), there exist... | Yes |
Corollary 17.1 Let \( E \) be a bounded, Lebesgue measurable set in \( {\mathbb{R}}^{N} \) and let \( \mathcal{F} \) be a fine Vitali covering for \( E \) . For every \( \varepsilon > 0 \), there exists a finite collection \( \left\{ {{Q}_{1},\ldots ,{Q}_{{n}_{\varepsilon }}}\right\} \) of cubes in \( \mathcal{F} \), w... | Proof Having fixed \( \varepsilon > 0 \), let \( {E}_{o,\varepsilon } \) be an open set containing \( E \) and satisfying (12.1). Introduce the subfamily \( {\mathcal{F}}_{\varepsilon } \) of the cubes in \( \mathcal{F} \) that are contained in \( {E}_{o,\varepsilon } \), and out of \( {\mathcal{F}}_{\varepsilon } \) s... | Yes |
Theorem 18.1 (Besicovitch [16]) Let \( E \) be a bounded subset of \( {\mathbb{R}}^{N} \) and let \( \mathcal{F} \) be a Besicovitch covering for \( E \) such that\n\n\[ \n\{ \text{the supremum of the radii of the balls in}\mathcal{F}\} \overset{\text{ def }}{ = }R < \infty \text{.}\n\]\n\n(18.1)\n\nThere exists a coun... | Proof Since \( E \) is bounded and \( R < \infty \), may assume that \( E \) and the balls making up the family \( \mathcal{F} \) are all included in some large ball \( {B}_{o} \) centered at the origin. Set \( {E}_{1} = E \) and\n\n\( {\mathcal{F}}_{1} = \left\{ \right. \) the collection of balls \( B\left( x\right) \... | Yes |
Proposition 18.1 There exists \( {c}_{N} \in \mathbb{N} \) depending only on \( N \) and independent of \( E \) , such that, for every \( k \in \mathbb{N} \), at most \( {c}_{N} \) balls out of \( \left\{ {{B}_{1},\ldots ,{B}_{k}}\right\} \) intersect \( {B}_{k} \) . | ## 19 Proof of Proposition 18.1\n\nFix some positive integer \( k \), consider those balls \( {B}_{j} \) for \( j = 1,\ldots, k \) that intersect \( {B}_{k} = {B}_{{\rho }_{k}}\left( {x}_{k}\right) \), and divide them into two sets\n\n\[{\mathcal{G}}_{1} = \left\{ \begin{matrix} \text{ the collection of balls }{B}_{j} ... | Yes |
Lemma 19.1 The number of balls in \( {\mathcal{G}}_{1} \) does not exceed \( {4}^{N}{\left( M + 1\right) }^{N} \) . | Proof Let \( \# \left\{ {\mathcal{G}}_{1}\right\} \) be the number of elements in \( {\mathcal{G}}_{1} \) . The balls \( \left\{ {{B}_{\frac{1}{3}{\rho }_{j}}\left( {x}_{j}\right) }\right\} \) are disjoint and are contained in \( {B}_{\left( {M + 1}\right) {\rho }_{k}}\left( {x}_{k}\right) \) . Indeed, since \( {B}_{j}... | Yes |
Lemma 19.2 The number \( M \) can chosen so that \( \theta > {\theta }_{o} = \arccos \frac{5}{6} \) . | Proof Assume \( n < m < k \) . By construction \( {x}_{m} \notin {B}_{{\rho }_{n}}\left( {x}_{n}\right) \), i.e.,\n\n\[ \left| {{x}_{n} - {x}_{m}}\right| > {\rho }_{n} \]\n\n(19.1)\n\nAlso \( {x}_{k} \notin {B}_{{\rho }_{n}}\left( {x}_{n}\right) \cup {B}_{{\rho }_{m}}\left( {x}_{m}\right) \), i.e.,\n\n\[ {\rho }_{n} < ... | Yes |
Theorem 20.1 (Besicovitch [16]) Let \( E \) be a bounded set in \( {\mathbb{R}}^{N} \) and let \( \mathcal{F} \) be a fine Besicovitch covering for \( E \) . Let \( \mu \) be a Radon measure in \( {\mathbb{R}}^{N} \) and let \( {\mu }_{e} \) be the outer measure associated with it.\n\nThere exists a countable collectio... | Proof May assumes \( {\mu }_{e}\left( E\right) > 0 \) otherwise the statement is trivial. Since \( E \) is bounded, may assume that both \( E \) and all the balls making up the covering \( \mathcal{F} \) are contained in some larger ball \( {B}_{o} \) .\n\nLet \( {\mathcal{B}}_{j} \) for \( j = 1,\ldots ,{c}_{N} \) be ... | Yes |
Proposition 1.1 A function \( f : E \rightarrow {\mathbb{R}}^{ * } \) is measurable if and only if at least one of the sets in (1.1) is measurable for all \( c \in \mathbb{Q} \) . | Proof Assume for example that the third of the sets in (1.1) is measurable for all \( c \in \mathbb{Q} \) . Having fixed some \( c \in \mathbb{R} - \mathbb{Q} \) let \( \left\{ {q}_{n}\right\} \) be a sequence of rational numbers decreasing to \( c \) . Then \( \left\lbrack {f > c}\right\rbrack = \bigcup \left\lbrack {... | No |
Proposition 1.2 Let \( f : E \rightarrow {\mathbb{R}}^{ * } \) be measurable and let \( \alpha \in \mathbb{R} - \{ 0\} \) . Then\n\n(i) The functions \( \left| f\right| ,{\alpha f},\alpha + f,{f}^{2} \), are measurable\n\n(ii) If \( f \neq 0 \) then also \( 1/f \) is measurable\n\n(iii) For any measurable subset \( {E}... | Proof The statements (i) and (ii) follow from the set identities:\n\n\[ \left\lbrack {\left| f\right| > c}\right\rbrack = \left\{ \begin{array}{ll} \left\lbrack {f > c}\right\rbrack \cup \left\lbrack {f < - c}\right\rbrack & \text{ if }c \geq 0 \\ E & \text{ if }c < 0 \end{array}\right. \]\n\n\[ \left\lbrack {\alpha + ... | Yes |
Proposition 1.3 Let \( f : E \rightarrow {\mathbb{R}}^{ * } \) and \( g : E \rightarrow \mathbb{R} \) be measurable. Then\n\n(i) the set \( \left\lbrack {f > g}\right\rbrack \) is measurable\n\n(ii) the functions \( f \pm g \) are measurable\n\n(iii) the function \( {fg} \) is measurable\n\n(iv) if \( g \neq 0 \), the ... | Proof Let \( \left\{ {q}_{n}\right\} \) be the sequence of the rational numbers. Then\n\n\[ \left\lbrack {f > g}\right\rbrack = \bigcup \left\lbrack {f \geq {q}_{n}}\right\rbrack \cap \left\lbrack {g < {q}_{n}}\right\rbrack . \]\n\nThis proves (i). To prove (ii) observe that for all \( c \in \mathbb{R} \)\n\n\[ \left\l... | Yes |
Proposition 1.4 Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of measurable functions in \( E \) . Then the functions\n\n\[ \n\\varphi = \\sup {f}_{n},\\;\\psi = \\inf {f}_{n},\\;{f}^{\\prime \\prime } = \\limsup {f}_{n},\\;{f}^{\\prime } = \\liminf {f}_{n}\n\]\n\nare measurable. | Proof For every \( c \\in \\mathbb{R} \) ,\n\n\[ \n\\left\\lbrack {\\varphi > c}\\right\\rbrack = \\bigcup \\left\\lbrack {{f}_{n} > c}\\right\\rbrack \\;\\text{ and }\\;\\left\\lbrack {\\psi \\geq c}\\right\\rbrack = \\bigcap \\left\\lbrack {{f}_{n} \\geq c}\\right\\rbrack .\n\] | Yes |
Proposition 2.1 Let \( \left\{ {f}_{n}\right\} \) be a sequence of measurable functions defined on a measurable set \( E \in \mathcal{A} \) of finite measure, and with values in \( {\mathbb{R}}^{ * } \). Assume that, for example, the second(first) of (2.2) holds. Then for every \( \eta > 0 \) there exists a measurable ... | Proof The statement is only proved for the lower limit, the arguments for the upper limit being analogous. Fix \( m, n \in \mathbb{N} \), and introduce the sets \[ {E}_{m, n} = \mathop{\bigcap }\limits_{{j = n}}^{\infty }\left\{ {x \in \left( {E - {\mathcal{E}}^{\prime }}\right) \mid {f}_{j}\left( x\right) \geq {f}^{\p... | Yes |
Proposition 2.2 Let \( \{ X,\mathcal{A},\mu \} \) be \( {\mathbb{R}}^{N} \) endowed with a inner regular Borel measure \( \mu \) (in the sense of (16.2) of Chap. 2). Let \( \left\{ {f}_{n}\right\} \) be a sequence of \( \mu \) measurable functions defined on a \( \mu \) -measurable set \( E \subset {\mathbb{R}}^{N} \),... | Proof The set \( {E}_{\eta } \) in (2.3) is a \( \mu \) -measurable subset of \( {\mathbb{R}}^{N} \) of finite measure. Since \( \mu \) is inner regular, there exists a closed set \( {E}_{c,\eta } \subset {E}_{\eta } \) such that \( \mu \left( {{E}_{\eta } - {E}_{c,\eta }}\right) \leq \eta \) . | Yes |
Proposition 3.1 Let \( f : E \rightarrow {\mathbb{R}}^{ * } \) be a nonnegative measurable function. There exists a sequence of simple functions \( \left\{ {f}_{n}\right\} \), such that \( {f}_{n} \leq {f}_{n + 1} \) and\n\n\[ f\left( x\right) = \lim {f}_{n}\left( x\right) \;\text{ for all }x \in E. \] | Proof For a fixed \( n \in \mathbb{N} \) set\n\n\[ {f}_{n}\left( x\right) = \left\{ \begin{array}{ll} n & \text{ if }f\left( x\right) \geq n \\ \frac{j}{{2}^{n}} & \text{ if }\frac{j}{{2}^{n}} \leq f\left( x\right) < \frac{j + 1}{{2}^{n}} \\ & \text{ for }j = 0,1,\ldots, n{2}^{n} - 1. \end{array}\right. \]\n\n(3.3)\n\n... | Yes |
Proposition 4.1 The convergence in measure identifies the limit uniquely up to a set of measure zero, i.e., if \( \left\{ {f}_{n}\right\} \) converges in measure to \( f \) and \( g \), then \( f = g \) a.e. in \( E \) . | Proof For \( n \in \mathbb{N} \) and a.e. \( x \in E \) ,\n\n\[ \left| {f\left( x\right) - g\left( x\right) }\right| \leq \left| {f\left( x\right) - {f}_{n}\left( x\right) }\right| + \left| {{f}_{n}\left( x\right) - g\left( x\right) }\right| . \]\n\nTherefore, for all \( \eta > 0 \)\n\n\[ \left\lbrack {\left| {f - g}\r... | Yes |
Proposition 4.2 Let \( \{ X,\mathcal{A},\mu \} \) be a measure space and let \( E \in \mathcal{A} \) be of finite measure. If \( \left\{ {f}_{n}\right\} \) converges a.e. in \( E \) to a function \( f : E \rightarrow {\mathbb{R}}^{ * } \) which is finite a.e. in \( E \), then \( \left\{ {f}_{n}\right\} \) converges to ... | Proof Having fixed an arbitrary \( \varepsilon > 0 \), by the Egorov-Severini theorem, there exists a measurable set \( {E}_{\varepsilon } \subset E \), such that \( \mu \left( {E - {E}_{\varepsilon }}\right) \leq \varepsilon \) and \( \left\{ {f}_{n}\right\} \) converges to \( f \) uniformly in \( {E}_{\varepsilon } \... | Yes |
Proposition 4.3 (Riesz [126]) Let \( \{ X,\mathcal{A},\mu \} \) be a measure space and let \( E \in \mathcal{A} \) . Let \( \left\{ {f}_{n}\right\} \) and \( f \) be measurable functions from \( E \) into \( {\mathbb{R}}^{ * } \), a.e. finite in \( E \) . If \( \left\{ {f}_{n}\right\} \) converges in measure to \( f \)... | Proof For \( m, n \in \mathbb{N} \), arguing as in Proposition 4.1\n\n\[ \mu \left( \left\lbrack {\left| {{f}_{n} - {f}_{m}}\right| > \eta }\right\rbrack \right) \leq \mu \left( \left\lbrack {\left| {{f}_{n} - f}\right| > \frac{1}{2}\eta }\right\rbrack \right) + \mu \left( \left\lbrack {\left| {{f}_{m} - f}\right| > \f... | Yes |
Proposition 4.4 Let \( \{ X,\mathcal{A},\mu \} \) be a measure space and let \( E \in \mathcal{A} \) be of finite measure. Let \( \left\{ {f}_{n}\right\} \) be a sequence of measurable functions from a measurable set \( E \) of finite measure, into \( {\mathbb{R}}^{ * } \) . The sequence \( \left\{ {f}_{n}\right\} \) c... | Proof The necessary condition has been established in the first part of the proof of Proposition 4.3, leading to (4.2). To prove its sufficiency, let (4.2) hold for all \( \eta > 0 \) and let \( \left\{ {f}_{{n}^{\prime }}\right\} \) be a subsequence, selected out of \( \left\{ {f}_{n}\right\} \) and convergent a.e. in... | Yes |
Proposition 5.1 A simple function defined in a measurable set \( E \subset {\mathbb{R}}^{N} \) of finite measure is quasi-continuous. | Proof Let \( f : E \rightarrow \mathbb{R} \) be simple and let \( \left\{ {{f}_{1},\ldots ,{f}_{n}}\right\} \) be its range. Since the sets \( {E}_{i} \) , defined in (3.1) are measurable, having fixed \( \varepsilon > 0 \), there exists closed sets \( {E}_{c, i} \subset {E}_{i} \) such that\n\n\[ \mu \left( {{E}_{i} -... | Yes |
Lemma 8.1 (Fatou [43]) Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of measurable and a.e. nonnegative functions in E. Then\n\n\[ \n{\\int }_{E}\\liminf {f}_{n}{d\\mu } \\leq \\liminf {\\int }_{E}{f}_{n}{d\\mu }\n\] | Proof Set \( f = \\liminf {f}_{n} \) and select a nonnegative simple function \( \\zeta \\in {\\mathcal{S}}_{f} \) . Assume first that \( \\zeta \) be integrable, so that it vanishes outside a measurable set \( F \\subset E \), of finite measure. For fixed \( x \\in F \) and \( \\varepsilon > 0 \) there exists an index... | Yes |
Theorem 8.1 (Monotone Convergence) Let \( \\left\\{ {f}_{n}\\right\\} \) be a monotone increasing sequence of measurable, nonnegative functions in E, i.e.,\n\n\[ 0 \\leq {f}_{n}\\left( x\\right) \\leq {f}_{n + 1}\\left( x\\right) \\;\\text{ for all }x \\in E\\text{ and for all }n \\in \\mathbb{N}. \]\n\nThen\n\n\[ \\li... | Proof The sequence \( \\left\\{ {f}_{n}\\right\\} \) converges for all \( x \\in E \) to a measurable function \( f : E \\rightarrow \) \( {\\mathbb{R}}^{ * } \) . Therefore, by Fatou’s lemma\n\n\[ {\\int }_{E}{fd\\mu } = {\\int }_{E}\\lim {f}_{n}{d\\mu } \\leq \\lim {\\int }_{E}{f}_{n}{d\\mu } \\leq {\\int }_{E}{fd\\m... | Yes |
Proposition 9.1 Let \( f, g : E \rightarrow \mathbb{R} \) be integrable. Then for all \( \alpha ,\beta \in \mathbb{R} \)\n\n\[{\int }_{E}\left( {{\alpha f} + {\beta g}}\right) {d\mu } = \alpha {\int }_{E}{fd\mu } + \beta {\int }_{E}{gd\mu }.\]\n\n(9.1) | Proof For \( \alpha \geq 0 \) denote by \( \alpha {\mathcal{S}}_{f} \) the collection of functions of the form \( {\alpha \zeta } \) where \( \zeta \in {\mathcal{S}}_{f} \) . If \( \alpha \geq 0 \) and \( f \geq 0 \) then \( \alpha {\mathcal{S}}_{f} = {\mathcal{S}}_{\alpha f} \) . Therefore,\n\n\[{\int }_{E}{\alpha fd\... | Yes |
Corollary 9.1 Let \( f : E \rightarrow {\mathbb{R}}^{ * } \) be integrable and let \( E \) be of finite measure. Then | \[ \mu \left( E\right) \mathop{\inf }\limits_{E}f \leq {\int }_{E}{fd\mu } \leq \mu \left( E\right) \mathop{\sup }\limits_{E}f. \] | Yes |
Proposition 10.1 Let \( g : E \rightarrow {\mathbb{R}}^{ * } \) be integrable and assume that \( {f}_{n} \geq g \) a.e. in \( E \) for all \( n \in \mathbb{N} \) . Then\n\n\[ \lim \inf {\int }_{E}{f}_{n}{d\mu } \geq {\int }_{E}\lim \inf {f}_{n}{d\mu }. \] | Proof Since \( {f}_{n} - g \geq 0 \), the sequence \( \left\{ {{f}_{n} - g}\right\} \) satisfies the assumptions of Fatou’s lemma. Thus\n\n\[ \lim \inf {\int }_{E}{f}_{n}{d\mu } \geq {\int }_{E}{gd\mu } + {\int }_{E}\left( {\lim \inf {f}_{n} - g}\right) {d\mu }. \] | Yes |
Proposition 10.2 Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of nonnegative, measurable, functions on E. Then\n\[ \n\\sum {\\int }_{E}{f}_{n}{d\\mu } = {\\int }_{E}\\sum {f}_{n}{d\\mu }\n\] | Proof The sequence \( \\left\\{ {\\mathop{\\sum }\\limits_{{i = 1}}^{n}{f}_{i}}\\right\\} \) is a monotone sequence of nonnegative, measurable functions. | No |
Theorem 10.1 (Dominated Convergence) Let \( \\left\\{ {f}_{n}\\right\\} \) be a dominated and convergent sequence of integrable functions in \( E \), i.e.,\n\n\[ \n\\lim {f}_{n}\\left( x\\right) = f\\left( x\\right) \\;\\text{ for all }x \\in E \n\]\n\nand there exists an integrable function \( g : E \\rightarrow {\\ma... | Proof The limit function \( f \) is measurable and by Fatou’s lemma\n\n\[ \n{\\int }_{E}\\left| f\\right| {d\\mu } \\leq \\lim \\inf {\\int }_{E}\\left| {f}_{n}\\right| {d\\mu } \\leq {\\int }_{E}{gd\\mu } < \\infty .\n\]\n\nThus \( f \) is integrable. Next\n\n\[ \ng - {f}_{n} \\geq 0\\;\\text{ and }\;{f}_{n} + g \\geq... | Yes |
Theorem 11.1 (Vitali [169]) Let \( E \) be measurable and let \( f : E \rightarrow {\mathbb{R}}^{ * } \) be integrable. For every \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that for every measurable subset \( \mathcal{E} \subset E \) of measure less than \( \delta \)\n\n\[{\int }_{\varepsilon }\left| f\ri... | Proof May assume that \( f \geq 0 \) . For \( n \in \mathbb{N} \) consider the functions\n\n\[E \ni x \rightarrow {f}_{n}\left( x\right) = \left\{ \begin{array}{ll} f\left( x\right) & \text{ if }f\left( x\right) < n \\ n & \text{ if }f\left( x\right) \geq n. \end{array}\right.\]\n\nSince \( \left\{ {f}_{n}\right\} \) i... | Yes |
Proposition 12.1 Let \( \\left\\{ {{A}_{n} \\times {B}_{n}}\\right\\} \) be a countable collection of disjoint, measurable rectangles whose union is a measurable rectangle \( A \\times B \) . Then\n\n\[ \n\\lambda \\left( {A \\times B}\\right) = \\sum \\lambda \\left( {{A}_{n} \\times {B}_{n}}\\right) .\n\] | Proof For each \( x \\in A \)\n\n\[ \nB = \\bigcup \\left\\{ {{B}_{j} \\mid \\left( {x, y}\\right) \\in {A}_{j} \\times {B}_{j};y \\in B}\\right\\} .\n\]\n\nSince for each \( x \\in A \) fixed this is a disjoint union\n\n\[ \n\\nu \\left( B\\right) {\\chi }_{A}\\left( x\\right) = \\sum \\nu \\left( {B}_{j}\\right) {\\c... | Yes |
Proposition 13.1 Let \( E \in {\left( \mathcal{A} \times \mathcal{B}\right) }_{o} \) . Then for every \( y \in Y \) the \( Y \) -section \( {E}_{y} \) is in \( \mathcal{A} \) and for every \( x \in X \) the \( X \) -section \( {E}_{x} \) is in \( \mathcal{B} \) . | Proof The collection \( \mathcal{F} \) of all sets \( E \in \left( {\mathcal{A} \times \mathcal{B}}\right) \) such that \( {E}_{x} \in \mathcal{B} \) for all \( x \in X \) is a \( \sigma \) -algebra. Since \( \mathcal{F} \) contains all the measurable rectangles, it must contain the smallest \( \sigma \) -algebra gener... | No |
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