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If $F$ is a sup-continuous function from the set of measurable functions to itself, then the least fixed point of $F$ is measurable. |
If $F$ is an inf-continuous function and $P$ is a predicate such that $P(M,s)$ implies that $F(A,s)$ is measurable whenever $A$ is measurable, then the greatest fixed point of $F$ is measurable. |
If $f$ is a measurable function from a measurable space $X$ to the extended natural numbers, then $f$ is measurable. |
If $P$ is a measurable predicate on a countable index set $I$, then the function $x \mapsto \text{the } i \text{ such that } P(i, x)$ is measurable. |
If $P_i$ is a countable collection of measurable predicates, then the predicate $\exists!i \in I. P_i$ is measurable. |
If $F$ is a measurable function from a measurable space $X$ to the set of natural numbers, then the function $x \mapsto \sup F(x)$ is measurable. |
If $f$ is measurable if $c$ is true, and $g$ is measurable if $c$ is false, then $f$ is measurable if $c$ is true, and $g$ is measurable if $c$ is false. |
If $S$ is a measurable set, then a predicate $P$ is measurable on the restriction of $M$ to $S$ if and only if $P$ is measurable on $M$ and $P$ is true on $S$. |
If $T$ is a measurable predicate, and $R$ is a measurable predicate-valued function, then $R^n(T)$ is a measurable predicate. |
If $f$ is a measurable function from a measurable space $M$ to the countable space $\mathbb{N}$, and $Q_i$ is a measurable predicate on $M$ for each $i \in \mathbb{N}$, then the predicate $\{x \in M \mid f(x) = i \text{ and } Q_i(x)\}$ is measurable. |
If $A_n$ is a sequence of measurable sets, then $\limsup A_n$ is measurable. |
If $A_n$ is a sequence of measurable sets, then $\liminf A_n$ is measurable. |
If $f$ is a measurable function from a measurable space $M$ to the countable space $\mathbb{N} \cup \{\infty\}$, and $g$ and $h$ are measurable functions from $M$ to a measurable space $N$, then the function $x \mapsto g(x)$ if $f(x) \in \mathbb{N}$ and $h(x)$ if $f(x) = \infty$ is measurable. |
If $f$ is a function from the natural numbers to the extended nonnegative reals and $A$ is a disjoint family of sets, then $\sum_{n=1}^\infty f(n) \cdot \mathbb{1}_{A_n}(x) = f(i)$ if $x \in A_i$. |
If the family $A$ is disjoint, then the sum of the indicator functions of the sets in $A$ is equal to the indicator function of the union of the sets in $A$. |
If $A$ is a disjoint family of sets, then for any $x \in A_j$, we have $\sum_{i \in P} f_i \cdot \mathbf{1}_{A_i}(x) = f_j$. |
If $f$ is a function that maps the empty set to $0$, then the sum of $f$ over the binary sets of $A$ and $B$ converges to $f(A) + f(B)$. |
If $f$ is a function that satisfies $f(\emptyset) = 0$, then the sum of $f$ over the set of binary strings of length $n$ is equal to the sum of $f$ over the set of binary strings of length $n$ that begin with $0$ plus the sum of $f$ over the set of binary strings of length $n$ that begin with $1$. |
If $f$ is a function from sets to a commutative monoid with identity $0$, then $\sum_{n=0}^\infty f(A \cup B^n) = f(A) + f(B)$. |
If $f$ is subadditive, then $f(x \cup y) \leq f(x) + f(y)$ for any disjoint sets $x$ and $y$. |
If $f$ is a positive countably subadditive function, then it is subadditive. |
If $f$ is a positive measure, then $f(\emptyset) = 0$. |
If $f$ is a positive measure, then $f(\emptyset) = 0$. |
If $f$ is additive, then $f(x \cup y) = f(x) + f(y)$ for any disjoint sets $x$ and $y$. |
If $f$ is an increasing function on a set $M$, then for any $x,y \in M$ such that $x \subseteq y$, we have $f(x) \leq f(y)$. |
If $f$ is a function from a $\sigma$-algebra $M$ to the extended real numbers such that for any countable family of disjoint sets in $M$, the sum of $f$ over the family is equal to $f$ of the union of the family, then $f$ is countably additive. |
If $f$ is a positive additive function on a ring of sets, and $A$ is an increasing sequence of sets in the ring, then $\sum_{i \leq n} f(A_i \setminus A_{i-1}) = f(A_n)$. |
If $f$ is a positive additive function on a ring of sets, then $f$ is a measure. |
If $f$ is a positive additive function on a ring of sets, then $f$ is increasing. |
Suppose $f$ is a positive additive function defined on a ring of sets $M$. If $A$ is a function from a finite set $S$ to $M$, then $f(\bigcup_{i \in S} A_i) \leq \sum_{i \in S} f(A_i)$. |
If $f$ is a positive countably additive function, then it is additive. |
If $f$ is a positive, additive, and increasing function on a measure space, then the sum of $f$ over a disjoint family of sets is bounded by $f$ of the whole space. |
If $\mu$ is a positive additive measure on a finite set $\Omega$, then $\mu$ is countably additive. |
A function $f$ is countably additive if and only if it is continuous from below. |
Let $f$ be a positive additive set function on a ring of sets $M$. Then $f$ is continuous from above if and only if $f$ is continuous from above on the empty set. |
If $f$ is a positive, additive function on a ring of sets $M$ such that $f(A) \neq \infty$ for all $A \in M$, and if $f$ is continuous from below in the sense that if $A_n$ is a decreasing sequence of sets in $M$ with empty intersection, then $\lim_{n \to \infty} f(A_n) = 0$, then $f$ is continuous from above in the se... |
If $f$ is a positive additive function on a ring of sets $M$ such that $f(A) \neq \infty$ for all $A \in M$, and if $f$ is continuous from below, then $f$ is countably additive. |
The measure of any set is non-negative. |
The measure of the empty set is zero. |
If the measure of a singleton set is nonzero, then the singleton set is in the space. |
The measure $\mu$ is countably additive. |
If $A$ is a disjoint family of measurable sets, then the sum of the measures of the sets in $A$ is equal to the measure of the union of the sets in $A$. |
If $F$ is a disjoint family of measurable sets, then the sum of the measures of the sets in $F$ is equal to the measure of the union of the sets in $F$. |
The measure of a disjoint union of sets is the sum of the measures of the sets. |
If $A$ and $B$ are disjoint measurable sets, then $\mu(A \cup B) = \mu(A) + \mu(B)$. |
If $A$ and $B$ are measurable sets, then the measure of their union is the sum of the measures of $A$ and $B$ minus the measure of their intersection. |
If $A$ and $B$ are measurable sets, then $\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B)$. |
If $F$ is a finite family of disjoint measurable sets, then the sum of the measures of the sets in $F$ is equal to the measure of the union of the sets in $F$. |
If $a \subseteq b$ and $b$ is measurable, then $\mu(a) \leq \mu(b)$. |
The measure of any set is less than or equal to the measure of the whole space. |
If $B$ is a finite measure subset of $A$, then the measure of $A - B$ is the measure of $A$ minus the measure of $B$. |
If $s$ is a measurable set and $s$ has finite measure, then the measure of the complement of $s$ is the total measure of the space minus the measure of $s$. |
If $A_1 \subseteq A_2 \subseteq \cdots$ is an increasing sequence of sets, then $\mu(A_1) \leq \mu(A_2) \leq \cdots$ and $\lim_{n \to \infty} \mu(A_n) = \mu(\bigcup_{n=1}^\infty A_n)$. |
If $B_1 \subseteq B_2 \subseteq \cdots$ is a sequence of sets, then $\mu(B_1) \leq \mu(B_2) \leq \cdots$. |
If $A_1 \subseteq A_2 \subseteq \cdots$ is an increasing sequence of sets, then $\mu(\bigcup_{i=1}^\infty A_i) = \lim_{n \to \infty} \mu(A_n)$. |
If $B_1, B_2, \ldots$ is a decreasing sequence of sets, then $\mu(B_1), \mu(B_2), \ldots$ is a decreasing sequence of numbers. |
If $A_1, A_2, \ldots$ is a decreasing sequence of sets in a measure space, then the infimum of the measures of the $A_i$ is equal to the measure of the intersection of the $A_i$. |
If $A_1, A_2, \ldots$ is a decreasing sequence of sets in a measure space, and if the measure of at least one of the sets is finite, then the infimum of the measures of the sets is equal to the measure of their intersection. |
If $F_i$ is a decreasing sequence of sets, then $\mu(\bigcap_i F_i) = \inf_i \mu(F_i)$. |
If $A_1, A_2, \ldots$ is a decreasing sequence of sets of finite measure, then the measure of their intersection is the limit of the measures of the sets. |
If $F$ is a sup-continuous operator on the set of measurable predicates on a measure space $M$, then the measure of the set of points $x$ such that the least fixed point of $F$ at $x$ is true is equal to the supremum of the measures of the sets of points $x$ such that the $i$th iterate of $F$ at $x$ is true. |
If $F$ and $f$ are sup-continuous, $F$ is measurable, and $f$ satisfies a certain condition, then $\mu(\{x \in X : \text{lfp}(F)(x)\}) = \text{lfp}(f)(s)$. |
If $I$ is a finite set and $A_i \in \mathcal{M}$ for all $i \in I$, then $\mu(\bigcup_{i \in I} A_i) \leq \sum_{i \in I} \mu(A_i)$. |
If $A$ and $B$ are measurable sets, then the measure of $A \cup B$ is less than or equal to the sum of the measures of $A$ and $B$. |
If $f$ is a function from the natural numbers to the measurable sets of a measure space, then the measure of the union of the sets in the range of $f$ is less than or equal to the sum of the measures of the sets in the range of $f$. |
If $x$ is an element of a measurable set $A$, then the measure of the set $\{x\} \cup A$ is the sum of the measures of $\{x\}$ and $A$. |
If $A$ is a non-empty set in a measure space $(X, \mathcal{M}, \mu)$, and $x \in X$ is not in $A$, then $\mu(A \cup \{x\}) = \mu(\{x\}) + \mu(A)$. |
If $S$ is a finite set and each element of $S$ is measurable, then the measure of $S$ is the sum of the measures of its elements. |
If $A$ is a subset of the union of a finite collection of disjoint sets, then the measure of $A$ is the sum of the measures of the intersections of $A$ with each of the sets in the collection. |
If $N$ is a set of measure zero, then any subset of $N$ also has measure zero. |
If the measure of each $N_i$ is zero, then the measure of the union of the $N_i$ is zero. |
If two finite measures agree on singletons, then they are equal. |
If two measures agree on a generator of the $\sigma$-algebra, then they are equal. |
If the underlying set of a measure space is empty, then the measure space is the empty measure space. |
Suppose $M$ and $N$ are two measures on a set $\Omega$ such that for every $X \in E$, we have $M(X) = N(X)$. If $E$ is a countable generator of the $\sigma$-algebra of $M$ and $N$, then $M = N$. |
The measure of a measure space is the measure space itself. |
If $A$ is a null set, then $\mu(A) = 0$. |
If $A$ is a null set, then $A$ is a measurable set. |
If $A$ is a measurable set with measure $0$, then $A$ is a null set. |
If $I$ is a countable nonempty set, then $\bigcup_{i \in I} N_i = \bigcup_{i \in \mathbb{N}} N_{f(i)}$, where $f$ is a bijection from $\mathbb{N}$ to $I$. |
If $I$ is a countable set and $N_i$ is a null set for each $i \in I$, then $\bigcup_{i \in I} N_i$ is a null set. |
If $N_i$ is a null set for each $i$, then $\bigcup_i N_i$ is a null set. |
If $B$ is a null set and $A$ is any set, then $A \cap B$ is a null set. |
If $B$ is a null set and $A$ is any set, then $B \cap A$ is a null set. |
If $B$ is a null set and $A$ is any set, then the measure of $A - B$ is the same as the measure of $A$. |
If $B$ is a null set and $A$ is any set, then $B - A$ is a null set. |
If $A$ is a measurable set and $B$ is a null set, then $A \cup B$ is measurable and has the same measure as $A$. |
If $A$ and $B$ are measurable sets and $A \cap B$ is a null set, then $A \cup B$ is measurable and $\mu(A \cup B) = \mu(A) + \mu(B)$. |
A property holds almost everywhere if and only if it holds outside a set of measure zero. |
If $N$ is a null set and $\{x \in X : \lnot P(x)\} \subseteq N$, then $P$ holds almost everywhere. |
A property holds almost everywhere if and only if the set of points where it fails is a null set. |
A set $N$ is a null set if and only if almost every point of $M$ is not in $N$. |
If $N$ is a null set, then almost every point of $M$ is not in $N$. |
If $N$ is a measurable set and $N$ is the set of points $x$ in the measure space $(X, \mathcal{A}, \mu)$ such that $P(x)$ is false, then $P(x)$ holds almost everywhere if and only if $\mu(N) = 0$. |
If $P$ is an event with probability $1$, then there exists a set $N$ with probability $0$ such that $P$ is true outside of $N$. |
If $P$ is a property that holds almost everywhere, then the set of points where $P$ does not hold has measure zero. |
If $P$ holds almost everywhere, then there exists a set $N$ of measure zero such that $P$ holds outside of $N$. |
If $N$ is a set of measure zero, then almost every element of $N$ satisfies the property $P$. |
If $P$ holds almost everywhere, and $P \implies Q$ holds almost everywhere, then $Q$ holds almost everywhere. |
If $P$ and $Q$ are two properties of a measure space $M$, then the following are equivalent: 1. Almost every element of $M$ satisfies $P$ and $Q$. 2. Almost every element of $M$ satisfies $P$. 3. Almost every element of $M$ satisfies $Q$. |
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