Proof of Basic Properties of a Measure

Throughout, $(X,\mathcal{F},\mu)$ and the notation are as in the statement. By σ-Algebra and Measurable Space (i), $\mathcal{F}\subseteq\mathcal{P}(X)$, with $\mathcal{P}(X)$ the power set; a sequence in $\mathcal{F}$ is thus a sequence in $\mathcal{P}(X)$, and a sequence in $\mathcal{P}(X)$ whose terms lie in $\mathcal{F}$ is a sequence in $\mathcal{F}$ (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v)). For a sequence $(E_{n})$ in $\mathcal{F}$, $n\mapsto\mu(E_{n})$ is a sequence in $[0,\infty]$, the composite of $(E_{n})$ with $\mu$ (Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (ii)); this applies to $(\mu(E_{n}))$, $(\mu(A_{m}))$, $(\mu(B_{m}))$ and $(\mu(D_{m}))$ below. Countable additivity and $\mu(\emptyset)=0$ are those of Measure, Measure Space, and Probability Measure (i).

Finite additivity. Let $r$ and $(A_{i})_{i\in[r]}$ be as in (i). By Maps Defined by Cases (i), applied on $\mathbb{N}$ to the condition $n\in[r]$ with the values $A_{n}\in\mathcal{F}$ and $\emptyset\in\mathcal{F}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vi)), there is a sequence $(E_{n})$ in $\mathcal{F}$ with $E_{n}=A_{n}$ for $n\in[r]$ and $E_{n}=\emptyset$ for $n\in\mathbb{N}\setminus[r]$. It is pairwise disjoint: for $m\neq k$ in $\mathbb{N}$, either $m,k\in[r]$ and $E_{m}\cap E_{k}=A_{m}\cap A_{k}=\emptyset$, or one of $E_{m},E_{k}$ is $\emptyset$. Its union is $\bigcup_{i=1}^{r}A_{i}$: an element of some $E_{n}$ shows $E_{n}\neq\emptyset$, so $n\in[r]$ and the element lies in $A_{n}$; and $A_{i}=E_{i}$ for $i\in[r]$. Hence $\bigcup_{i=1}^{r}A_{i}=\bigcup_{n\in\mathbb{N}}E_{n}\in\mathcal{F}$ by σ-Algebra and Measurable Space (i), and countable additivity gives $\mu\bigl(\bigcup_{i=1}^{r}A_{i}\bigr)=\sum_{n\in\mathbb{N}}\mu(E_{n})$. If $k\in\mathbb{N}$ and $r<k$, then $k\le r$ fails by Arithmetic and Order of the Natural Numbers (vi) and Arithmetic and Order of the Natural Numbers (vii), so $k\notin[r]$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i), and $\mu(E_{k})=\mu(\emptyset)=0$. By Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence (vii), $\sum_{n\in\mathbb{N}}\mu(E_{n})$ is therefore the $r$-th partial sum of $(\mu(E_{n}))$, the iterated sum of its restriction to $[r]$ (Iterated Operations: Finite Sums and Finite Products (ii)). That restriction is the map $i\mapsto\mu(A_{i})$ on $[r]$, as $E_{i}=A_{i}$ for $i\in[r]$, so the partial sum is $\sum_{i=1}^{r}\mu(A_{i})$.

Two sets. Let $A,B$ be as in (ii). By Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (iii), with $U=A$ and $V=B$, there is a sequence $(E_{n})$ in $\mathcal{P}(X)$ with $E_{1}=A$, $E_{2}=B$ and $E_{n}=\emptyset$ for $n\neq1,2$; its terms lie in $\mathcal{F}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vi)), so it is a sequence in $\mathcal{F}$. Since $2=1+1$ by Arithmetic and Order of the Natural Numbers (xiv), $1<2$ by Arithmetic and Order of the Natural Numbers (xi). The sequence is pairwise disjoint: for $m\neq k$, either $\{m,k\}=\{1,2\}$ and $E_{m}\cap E_{k}=A\cap B=\emptyset$, or one of $E_{m},E_{k}$ is $\emptyset$. Its union is $A\cup B$, since an element of some $E_{n}$ shows $E_{n}\neq\emptyset$, so $n=1$ or $n=2$. Countable additivity gives $\mu(A\cup B)=\sum_{n\in\mathbb{N}}\mu(E_{n})$. If $k\in\mathbb{N}$ and $2<k$, then $k\neq2$ by Arithmetic and Order of the Natural Numbers (vii), and $1<k$ by Arithmetic and Order of the Natural Numbers (vi), so $k\neq1$; thus $\mu(E_{k})=\mu(\emptyset)=0$. By Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence (vii) with $n=2$ and Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i) with $n=1$, $\sum_{n\in\mathbb{N}}\mu(E_{n})=\sum_{k=1}^{2}\mu(E_{k})=\mu(E_{1})+\mu(E_{2})=\mu(A)+\mu(B)$.

Monotonicity. Let $A\subseteq B$ in $\mathcal{F}$. Then $B\setminus A\in\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (x), $A\cap(B\setminus A)=\emptyset$, and $A\cup(B\setminus A)=B$ because $A\subseteq B$. By (ii), $\mu(B)=\mu(A)+\mu(B\setminus A)$, and $\mu(A)\le\mu(A)+\mu(B\setminus A)$ by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (v).

Differences. Let $A\subseteq B$ in $\mathcal{F}$ with $\mu(B)<\infty$. As in the previous paragraph, $\mu(B)=\mu(A)+\mu(B\setminus A)$. By (iii), $\mu(A)\le\mu(B)$ and, as $B\setminus A\subseteq B$, $\mu(B\setminus A)\le\mu(B)$; so $\mu(A)$ and $\mu(B\setminus A)$ are real (Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (iii)). By Addition and Multiplication on [0,∞] (i), $\mu(A)+\mu(B\setminus A)$ is then the sum in $\mathbb{R}$, so $\mu(B)$ is real and $\mu(B\setminus A)=\mu(B)-\mu(A)$ (Differences, Quotients and Absolute Values of Real Numbers, Read in the Type Real (i)). If $\mu$ is finite, then $\mu(X)<\infty$ by Measure, Measure Space, and Probability Measure (iii), $X\in\mathcal{F}$ by σ-Algebra and Measurable Space (i) and $A\subseteq X$ for $A\in\mathcal{F}$, so the case $B=X$ gives $\mu(X\setminus A)=\mu(X)-\mu(A)$, which is (v).

The disjointing sequence. This paragraph prepares the next ones. Let $(A_{n})$ be a sequence in $\mathcal{F}$. By Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (iv), there are sequences $(C_{n})$ and $(B_{n})$ in $\mathcal{P}(X)$ with $C_{1}=X$, $C_{n+1}=C_{n}\cap(X\setminus A_{n})$ and $B_{n}=A_{n}\cap C_{n}$ for every $n\in\mathbb{N}$. By The Disjointing Sequence of a Sequence of Sets, these sequences have the following properties. (D1) $(B_{n})$ is a sequence in $\mathcal{F}$, by The Disjointing Sequence of a Sequence of Sets (v), as every $A_{n}$ lies in $\mathcal{F}$. (D2) $B_{n}\subseteq A_{n}$ for every $n$, by The Disjointing Sequence of a Sequence of Sets (ii), and $(B_{n})$ is pairwise disjoint, $B_{m}\cap B_{k}=\emptyset$ for $m\neq k$, by The Disjointing Sequence of a Sequence of Sets (iii). (D3) If $n\in\mathbb{N}$ and $x\in A_{n}$, then $x\in B_{m}$ for some $m\in\mathbb{N}$ with $m\le n$, that is, for some $m\in[n]$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i); and $\bigcup_{n\in\mathbb{N}}B_{n}=\bigcup_{n\in\mathbb{N}}A_{n}$; both by The Disjointing Sequence of a Sequence of Sets (iv).

Countable subadditivity. Let $(A_{m})$ be a sequence in $\mathcal{F}$ and $(B_{m})$ its disjointing sequence as above. By (D3), (D1), (D2) and countable additivity, $\mu\bigl(\bigcup_{m}A_{m}\bigr)=\mu\bigl(\bigcup_{m}B_{m}\bigr)=\sum_{m\in\mathbb{N}}\mu(B_{m})$. By (D2) and (iii), $\mu(B_{m})\le\mu(A_{m})$ for every $m$, so $\sum_{m\in\mathbb{N}}\mu(B_{m})\le\sum_{m\in\mathbb{N}}\mu(A_{m})$ by Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence (x).

Subadditivity for two sets. Let $A,B\in\mathcal{F}$. Then $B\setminus A\in\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (x), $A\cap(B\setminus A)=\emptyset$ and $A\cup(B\setminus A)=A\cup B$, so $\mu(A\cup B)=\mu(A)+\mu(B\setminus A)$ by (ii). As $B\setminus A\subseteq B$, $\mu(B\setminus A)\le\mu(B)$ by (iii), and then $\mu(A)+\mu(B\setminus A)\le\mu(A)+\mu(B)$ by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (iv) and the commutativity in Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (i).

Continuity from below. Let $(A_{m})$ and $A$ be as in (viii), and $(B_{m})$ the disjointing sequence of $(A_{m})$. First, $A_{k}\subseteq A_{n}$ whenever $n\in\mathbb{N}$ and $k\in[n]$, as then $k\le n$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i) (Monotone Sequences of Sets, Unions of Traces, and Disjointness of Smaller Sets (i)). Now fix $n\in\mathbb{N}$. The family $(B_{k})_{k\in[n]}$, the restriction of $(B_{k})$ to $[n]$, is pairwise disjoint with terms in $\mathcal{F}$ by (D1) and (D2), and its union is $A_{n}$: for $k\in[n]$, $B_{k}\subseteq A_{k}\subseteq A_{n}$ by (D2) and the inclusions just shown, and every $x\in A_{n}$ lies in $B_{m}$ for some $m\in[n]$ by (D3). By (i) with $r=n$, $\mu(A_{n})=\sum_{k=1}^{n}\mu(B_{k})$, which is the $n$-th partial sum $s_{n}$ of $(\mu(B_{k}))$ (Iterated Operations: Finite Sums and Finite Products (ii)). Hence $\{s_{n}:n\in\mathbb{N}\}=\{\mu(A_{n}):n\in\mathbb{N}\}$, and by (D3), countable additivity and Sum of a Sequence in [0,∞] (ii),

$$\mu(A)=\mu\Bigl(\bigcup_{k\in\mathbb{N}}B_{k}\Bigr)=\sum_{k\in\mathbb{N}}\mu(B_{k})=\sup\{s_{n}:n\in\mathbb{N}\}=\sup_{n}\mu(A_{n}).$$

Continuity from above. Let $(A_{m})$ and $A$ be as in (ix); $A\in\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (viii). As $A_{1}\in\mathcal{F}\subseteq\mathcal{P}(X)$, every subset of $A_{1}$ is a subset of $X$, so $\mathcal{P}(A_{1})\subseteq\mathcal{P}(X)$. By Monotone Sequences of Sets, Unions of Traces, and Disjointness of Smaller Sets (ii), $A_{m}\subseteq A_{1}$ for every $m$; so $(A_{m})$, a sequence in $\mathcal{P}(X)$ whose terms lie in the subset $\mathcal{P}(A_{1})$ of $\mathcal{P}(X)$, is a sequence in $\mathcal{P}(A_{1})$ (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v)), and Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (i), applied to $A_{1}$, gives the sequence $(D_{m})$ in $\mathcal{P}(A_{1})$ with $D_{m}=A_{1}\setminus A_{m}$. As the range of $(D_{m})$ lies in $\mathcal{P}(A_{1})\subseteq\mathcal{P}(X)$, $(D_{m})$ is a sequence in $\mathcal{P}(X)$ by Functions, Values of a Function, and Functions from One Class to Another (iii). Its terms lie in $\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (x), so it is a sequence in $\mathcal{F}$ by Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v). Also $A\subseteq A_{m}$ for every $m$, by Indexed Families of Sets and Their Union, Intersection and Product (iii). As $\mu(A_{1})<\infty$, (iv) with $B=A_{1}$ shows that $\mu(A_{1})$, $\mu(A)$ and every $\mu(A_{m})$ are real, $\mu(D_{m})=\mu(A_{1})-\mu(A_{m})$ for every $m$, and $\mu(A_{1}\setminus A)=\mu(A_{1})-\mu(A)$. Moreover $D_{m}\subseteq D_{m+1}$, as $A_{m+1}\subseteq A_{m}$; and $\bigcup_{m}D_{m}=A_{1}\setminus A$, since $x$ lies in some $D_{m}$ exactly when $x\in A_{1}$ and $x\notin A_{m}$ for some $m$, that is, $x\in A_{1}$ and $x\notin\bigcap_{m}A_{m}$. By (viii) applied to $(D_{m})$,

$$\mu(A_{1})-\mu(A)=\mu(A_{1}\setminus A)=\sup_{m}\mu(D_{m}).$$

Let $S=\{\mu(A_{m}):m\in\mathbb{N}\}$. We show that $\mu(A)$ is the greatest lower bound of $S$, which makes it the infimum $\inf_{m}\mu(A_{m})$ by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv). It is a lower bound of $S$: $\mu(A)\le\mu(A_{m})$ by (iii), as $A\subseteq A_{m}$. Let $b\in[-\infty,\infty]$ be a lower bound of $S$. If $b=-\infty$, then $b\le\mu(A)$ by Order and Suprema in the Extended Real Line (ii). Otherwise $b\le\mu(A_{1})$ with $\mu(A_{1})$ real, so $b\neq\infty$ by Order and Suprema in the Extended Real Line (ii), and $b\in\mathbb{R}$ by The Extended Real Line, Its Order, and [0,∞] (ii). For every $m$, $b\le\mu(A_{m})$ holds in $\mathbb{R}$ by Order and Suprema in the Extended Real Line (i), so $\mu(D_{m})=\mu(A_{1})-\mu(A_{m})\le\mu(A_{1})-b$ (Differences, Quotients and Absolute Values of Real Numbers, Read in the Type Real (i)). Thus the real number $\mu(A_{1})-b$ is an upper bound of $\{\mu(D_{m}):m\in\mathbb{N}\}$, and the supremum, being the least upper bound (Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv)), satisfies $\mu(A_{1})-\mu(A)=\sup_{m}\mu(D_{m})\le\mu(A_{1})-b$. Both sides are real, so this holds in $\mathbb{R}$ by Order and Suprema in the Extended Real Line (i), and $b\le\mu(A)$ (Differences, Quotients and Absolute Values of Real Numbers, Read in the Type Real (i)). Hence $\mu(A)=\inf_{m}\mu(A_{m})$.