Proof of Extension of a Finite Premeasure from an Algebra of Sets to the Generated Sigma-Algebra

Each result cited is universally quantified over the data in its own statement.

Throughout, $\mathbb{N}=\{1,2,3,\dots\}$ as in Natural Numbers, and for a sequence $(a_{i})_{i\in\mathbb{N}}$ in $[0,\infty]$ the sum $\sum_{i\in\mathbb{N}}a_{i}$ is the one of Measure, Measure Space, and Probability Measure: if every $a_{i}$ is real and the partial sums $\sum_{i=1}^{n}a_{i}$ are bounded above, it is their least upper bound, and otherwise it is $\infty$. Two consequences of this definition are used repeatedly: if all $a_{i}$ are real and $c$ is a real number with $\sum_{i=1}^{n}a_{i}\le c$ for every $n\in\mathbb{N}$, then $\sum_{i\in\mathbb{N}}a_{i}\le c$; and if $\sum_{i\in\mathbb{N}}a_{i}$ is real, then every partial sum is at most $\sum_{i\in\mathbb{N}}a_{i}$. In particular, if $a_{i}=0$ for all $i$ outside a finite set $F\subseteq\mathbb{N}$, the partial sums are eventually constant equal to $\sum_{i\in F}a_{i}$, so $\sum_{i\in\mathbb{N}}a_{i}=\sum_{i\in F}a_{i}$. For a sequence $(a_{i})$ of nonnegative reals, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion shows that the series $\sum_{i=1}^{\infty}a_{i}$ of Series of Real Numbers §convergent converges exactly when the partial sums are bounded above, and then its sum is their least upper bound, that is, it equals $\sum_{i\in\mathbb{N}}a_{i}$.

Step 0: elementary properties of $\mathcal{A}$ and $m$. Let $A,B\in\mathcal{A}$. Then $A\cap B=\Omega\setminus\bigl((\Omega\setminus A)\cup(\Omega\setminus B)\bigr)\in\mathcal{A}$ and $A\setminus B=A\cap(\Omega\setminus B)\in\mathcal{A}$, and by induction on the number of sets every finite union of members of $\mathcal{A}$ belongs to $\mathcal{A}$, the empty union being $\varnothing\in\mathcal{A}$. If $A$ and $B$ are disjoint, the sequence $(C_{i})_{i\in\mathbb{N}}$ with $C_{1}=A$, $C_{2}=B$ and $C_{i}=\varnothing$ for $i\ge3$ consists of pairwise disjoint members of $\mathcal{A}$ with union $A\cup B\in\mathcal{A}$, so by the countable additivity of $m$ the series $\sum_{i=1}^{\infty}m(C_{i})$ converges to $m(A\cup B)$; since $m(\varnothing)=0$ its partial sums equal $m(A)+m(B)$ from $n=2$ on, so

$$m(A\cup B)=m(A)+m(B)\qquad(A,B\in\mathcal{A}\text{ disjoint}).$$

If $A\subseteq B$, then $B$ is the disjoint union of $A$ and $B\setminus A\in\mathcal{A}$, so $m(B)=m(A)+m(B\setminus A)\ge m(A)$. In particular $m(A)\le m(\Omega)$ for every $A\in\mathcal{A}$.

Step 1: the outer measure $m^{*}$. For $E\subseteq\Omega$, call a sequence $(A_{i})_{i\in\mathbb{N}}$ of members of $\mathcal{A}$ a cover of $E$ if $E\subseteq\bigcup_{i\in\mathbb{N}}A_{i}$, and call $\sum_{i\in\mathbb{N}}m(A_{i})\in[0,\infty]$ its weight. Let $W(E)$ be the set of those weights of covers of $E$ that are real numbers. The sequence $\Omega,\varnothing,\varnothing,\dots$ is a cover of $E$ of weight $m(\Omega)$, so $W(E)$ is a nonempty set of real numbers, and it is bounded below by $0$. Define $m^{*}(E)$ to be the greatest lower bound of $W(E)$. Then $m^{*}(E)$ is a real number with $0\le m^{*}(E)\le m(\Omega)$, and:

(a) every cover of $E$ has weight at least $m^{*}(E)$ (for a real weight because $m^{*}(E)$ is a lower bound of $W(E)$, for the weight $\infty$ by the conventions of Measure, Measure Space, and Probability Measure);

(b) for every real $\varepsilon>0$ there is a cover of $E$ with real weight less than $m^{*}(E)+\varepsilon$, because $m^{*}(E)+\varepsilon$ is not a lower bound of $W(E)$.

Step 2: $m^{*}$ is an outer measure on $\Omega$ in the sense of Outer Measure and Caratheodory Measurability. Its values lie in $[0,\infty)\subseteq[0,\infty]$.

Property 1. The sequence all of whose terms are $\varnothing$ is a cover of $\varnothing$ of weight $0$, so $0\le m^{*}(\varnothing)\le0$.

Property 2. If $E\subseteq F\subseteq\Omega$, every cover of $F$ is a cover of $E$, so $W(F)\subseteq W(E)$ and therefore $m^{*}(E)\le m^{*}(F)$.

Property 3. Let $(E_{k})_{k\in\mathbb{N}}$ be a sequence of subsets of $\Omega$ and put $E=\bigcup_{k\in\mathbb{N}}E_{k}$. If $\sum_{k\in\mathbb{N}}m^{*}(E_{k})=\infty$ there is nothing to prove, so assume that $s=\sum_{k\in\mathbb{N}}m^{*}(E_{k})$ is real; then $\sum_{k=1}^{K}m^{*}(E_{k})\le s$ for every $K\in\mathbb{N}$. Let a real $\varepsilon>0$ be given. For each $k\in\mathbb{N}$, by (b) choose a cover $(A_{k,i})_{i\in\mathbb{N}}$ of $E_{k}$ whose weight $w_{k}$ is real and satisfies $w_{k}<m^{*}(E_{k})+\varepsilon(1/2)^{k}$. The set $\mathbb{N}$ is countable by claim 1 of Basic Properties of Countable Sets, so $\mathbb{N}\times\mathbb{N}$ is countable by claim 1 of Products and Powers of Countable Sets; as it is nonempty, Countable Set provides a sequence $(\varphi(j))_{j\in\mathbb{N}}$ in $\mathbb{N}\times\mathbb{N}$ whose set of terms is $\mathbb{N}\times\mathbb{N}$. Call $j\in\mathbb{N}$ a first index if $\varphi(j')\ne\varphi(j)$ for every $j'\in\mathbb{N}$ with $j'<j$, and define $C_{j}=A_{\varphi(j)}$ if $j$ is a first index and $C_{j}=\varnothing$ otherwise; here $A_{(k,i)}$ means $A_{k,i}$. Each $C_{j}$ belongs to $\mathcal{A}$.

The sequence $(C_{j})_{j\in\mathbb{N}}$ is a cover of $E$: if $x\in E$, choose $k$ with $x\in E_{k}$ and $i$ with $x\in A_{k,i}$; the set of $j\in\mathbb{N}$ with $\varphi(j)=(k,i)$ is nonempty, and its least element $j_{0}$ is a first index, so $x\in A_{k,i}=C_{j_{0}}$.

Its weight is at most $s+\varepsilon$. Let $n\in\mathbb{N}$. If $j<j'\le n$ are both first indices, then $\varphi(j)\ne\varphi(j')$ by the definition of a first index applied to $j'$; so the pairs $\varphi(j)$, for the first indices $j\le n$, are pairwise distinct. Let $K$ and $I$ be the largest first, respectively second, coordinates of the finitely many pairs $\varphi(1),\dots,\varphi(n)$. Since all terms are nonnegative and distinct first indices contribute distinct pairs in $\{1,\dots,K\}\times\{1,\dots,I\}$,

$$\sum_{j=1}^{n}m(C_{j})\le\sum_{k=1}^{K}\sum_{i=1}^{I}m(A_{k,i})\le\sum_{k=1}^{K}w_{k}\le\sum_{k=1}^{K}m^{*}(E_{k})+\varepsilon\sum_{k=1}^{K}(1/2)^{k}\le s+\varepsilon ,$$

where the second inequality holds because each partial sum $\sum_{i=1}^{I}m(A_{k,i})$ is at most the real weight $w_{k}$, and the last because $\sum_{k=1}^{K}(1/2)^{k}\le1$: by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric the series $\sum_{k=1}^{\infty}(1/2)^{k}$ converges with sum $1$, and its partial sums are at most its sum by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. Hence every partial sum of $(m(C_{j}))_{j\in\mathbb{N}}$ is at most $s+\varepsilon$, so the weight of $(C_{j})$ is a real number at most $s+\varepsilon$, and by (a) $m^{*}(E)\le s+\varepsilon$. As $\varepsilon>0$ was arbitrary, $m^{*}(E)\le s$, which is property 3.

Step 3: $m^{*}(A)=m(A)$ for every $A\in\mathcal{A}$. The sequence $A,\varnothing,\varnothing,\dots$ is a cover of $A$ of weight $m(A)$, so $m^{*}(A)\le m(A)$. Conversely, let $(A_{i})_{i\in\mathbb{N}}$ be a cover of $A$ with real weight $w$. For $i\in\mathbb{N}$ let $U_{i}$ be the union of the sets $A_{j}$ with $j\in\mathbb{N}$ and $j<i$ (so $U_{1}=\varnothing$), and let $B_{i}=(A\cap A_{i})\setminus U_{i}$; by Step 0, $U_{i}$ and $B_{i}$ belong to $\mathcal{A}$. The sets $B_{i}$ are pairwise disjoint: if $i<i'$, then $B_{i}\subseteq A_{i}\subseteq U_{i'}$ while $B_{i'}\cap U_{i'}=\varnothing$. Their union is $A$: each $B_{i}\subseteq A$, and if $x\in A$, the set of $i$ with $x\in A_{i}$ is nonempty, and for its least element $i$ we have $x\notin U_{i}$, so $x\in B_{i}$. By the countable additivity of $m$, the series $\sum_{i=1}^{\infty}m(B_{i})$ converges with sum $m(A)$. Since $B_{i}\subseteq A_{i}$, Step 0 gives $m(B_{i})\le m(A_{i})$, so $\sum_{i=1}^{n}m(B_{i})\le\sum_{i=1}^{n}m(A_{i})\le w$ for every $n$; by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion, $m(A)$ is the least upper bound of these partial sums, so $m(A)\le w$. Thus $m(A)$ is a lower bound of $W(A)$, and $m(A)\le m^{*}(A)$.

Step 4: every $A\in\mathcal{A}$ is Carath'eodory measurable with respect to $m^{*}$. Let $A\in\mathcal{A}$ and $E\subseteq\Omega$. Applying property 3 of Step 2 to the sequence $E\cap A,\ E\setminus A,\ \varnothing,\varnothing,\dots$, whose union is $E$, and using $m^{*}(\varnothing)=0$, we get $m^{*}(E)\le m^{*}(E\cap A)+m^{*}(E\setminus A)$. For the reverse inequality let a real $\varepsilon>0$ be given and, by (b), choose a cover $(A_{i})_{i\in\mathbb{N}}$ of $E$ with real weight $w<m^{*}(E)+\varepsilon$. Then $(A_{i}\cap A)_{i\in\mathbb{N}}$ is a cover of $E\cap A$ and $(A_{i}\setminus A)_{i\in\mathbb{N}}$ is a cover of $E\setminus A$, all terms belonging to $\mathcal{A}$ by Step 0, and Step 0 also gives $m(A_{i})=m(A_{i}\cap A)+m(A_{i}\setminus A)$. Write $P_{n}=\sum_{i=1}^{n}m(A_{i}\cap A)$ and $Q_{n}=\sum_{i=1}^{n}m(A_{i}\setminus A)$; these are nondecreasing in $n$ and $P_{n}+Q_{n}=\sum_{i=1}^{n}m(A_{i})\le w$. For $n,n'\in\mathbb{N}$ and $N=\max\{n,n'\}$ we get $P_{n}+Q_{n'}\le P_{N}+Q_{N}\le w$; hence the weights $w'=\sup_{n}P_{n}$ and $w''=\sup_{n}Q_{n}$ of the two covers are real and $w'+w''\le w$. By (a),

$$m^{*}(E\cap A)+m^{*}(E\setminus A)\le w'+w''\le w<m^{*}(E)+\varepsilon .$$

As $\varepsilon>0$ was arbitrary, $m^{*}(E\cap A)+m^{*}(E\setminus A)\le m^{*}(E)$, and so $m^{*}(E)=m^{*}(E\cap A)+m^{*}(E\setminus A)$.

Step 5: existence (claim 1). Let $\mathcal{M}$ be the family of subsets of $\Omega$ that are Carath'eodory measurable with respect to $m^{*}$. By claim 1 of Caratheodory Extension Theorem, $\mathcal{M}$ is a $\sigma$-algebra on $\Omega$, and by Step 4 it contains $\mathcal{A}$; hence $\sigma(\mathcal{A})\subseteq\mathcal{M}$ by claim 2 of Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra. By claim 2 of Caratheodory Extension Theorem, the restriction of $m^{*}$ to $\mathcal{M}$ is a measure on $(\Omega,\mathcal{M})$. Let $\bar{m}$ be the restriction of $m^{*}$ to $\sigma(\mathcal{A})$. It takes values in $[0,\infty]$, satisfies $\bar{m}(\varnothing)=m^{*}(\varnothing)=0$, and every sequence of pairwise disjoint members of $\sigma(\mathcal{A})$ is a sequence of pairwise disjoint members of $\mathcal{M}$ whose union lies in the $\sigma$-algebra $\sigma(\mathcal{A})$; so the countable additivity of $m^{*}$ on $\mathcal{M}$ gives that of $\bar{m}$ on $\sigma(\mathcal{A})$, and $\bar{m}$ is a measure on $(\Omega,\sigma(\mathcal{A}))$ in the sense of Measure, Measure Space, and Probability Measure. By Step 3, $\bar{m}(A)=m^{*}(A)=m(A)$ for every $A\in\mathcal{A}$; in particular $\bar{m}(\Omega)=m(\Omega)$ is a real number, so $\bar{m}(\Omega)<\infty$ and $\bar{m}$ is finite.

Step 6: uniqueness (claim 2). The family $\mathcal{A}$ is a $\pi$-system in the sense of Dynkin's Pi-Lambda Theorem: it is nonempty since $\Omega\in\mathcal{A}$, and it is closed under finite intersections by Step 0. The $\sigma$-algebra $\sigma(\mathcal{A})$ of Generated Sigma-Algebra, the family of subsets of $\Omega$ belonging to every $\sigma$-algebra on $\Omega$ containing $\mathcal{A}$, is the intersection of all $\sigma$-algebras on $\Omega$ containing $\mathcal{A}$, which is the generated $\sigma$-algebra of Generated Sigma-Algebra. Let $\bar{m}_{1}$ and $\bar{m}_{2}$ be as in claim 2. They agree on $\mathcal{A}$, and $\bar{m}_{1}(\Omega)=\bar{m}_{2}(\Omega)=m(\Omega)<\infty$. Claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, applied to the measurable space $(\Omega,\sigma(\mathcal{A}))$, the $\pi$-system $\mathcal{A}$ and the measures $\bar{m}_{1}$ and $\bar{m}_{2}$, gives $\bar{m}_{1}=\bar{m}_{2}$.