# proof of Carathéodory’s extension theorem

The first step is to extend the set function $\mu_{0}$ to the power set $P(X)$. For any subset $S\subseteq X$ the value of $\mu^{*}(S)$ is defined by taking sequences $S_{i}$ in $A$ which cover $S$,

 $\mu^{*}(S)\equiv\inf\left\{\sum_{i=1}^{\infty}\mu_{0}(S_{i}):S_{i}\in A,\ S% \subseteq\bigcup_{i=1}^{\infty}S_{i}\right\}.$ (1)

We show that this is an outer measure (http://planetmath.org/OuterMeasure2). First, it is clearly non-negative. Secondly, if $S=\emptyset$ then we can take $S_{i}=\emptyset$ in (1) to obtain $\mu^{*}(S)\leq\sum_{i}\mu_{0}(\emptyset)=0$, giving $\mu^{*}(\emptyset)$=0. It is also clear that $\mu^{*}$ is increasing, so that if $S\subseteq T$ then $\mu^{*}(S)\leq\mu^{*}(T)$. The only remaining property to be proven is subadditivity. That is, if $S_{i}$ is a sequence in $P(X)$ then

 $\mu^{*}\left(\bigcup_{i}S_{i}\right)\leq\sum_{i}\mu^{*}(S_{i}).$ (2)

To prove this inequality, choose any $\epsilon>0$ and, by the definition (1) of $\mu^{*}$, for each $i$ there exists a sequence $S_{i,j}\in A$ such that $S_{i}\subseteq\bigcup_{j}S_{i,j}$ and,

 $\sum_{j=1}^{\infty}\mu_{0}(S_{i,j})\leq\mu^{*}(S_{i})+2^{-i}\epsilon.$

As $\bigcup_{i}S_{i}\subseteq\bigcup_{i,j}S_{i,j}$, equation (1) defining $\mu^{*}$ gives

 $\mu^{*}\left(\bigcup_{i}S_{i}\right)\leq\sum_{i,j}\mu_{0}(S_{i,j})=\sum_{i}% \sum_{j}\mu_{0}(S_{i,j})\leq\sum_{i}(\mu^{*}(S_{i})+2^{-i}\epsilon)=\sum_{i}% \mu^{*}(S_{i})+\epsilon.$

As $\epsilon>0$ is arbitrary, this proves subadditivity (2). So, $\mu^{*}$ is indeed an outer measure.

The next step is to show that $\mu^{*}$ agrees with $\mu_{0}$ on $A$. So, choose any $S\in A$. The inequality $\mu^{*}(S)\leq\mu_{0}(S)$ follows from taking $S_{1}=S$ and $S_{i}=\emptyset$ in (1), and it remains to prove the reverse inequality. So, let $S_{i}$ be a sequence in $A$ covering $S$, and set

 $S^{\prime}_{i}=(S\cap S_{i})\setminus\bigcup_{j=1}^{i-1}S_{j}\in A.$

Then, $S^{\prime}_{i}$ are disjoint sets satisfying $\bigcup_{j=1}^{i}S^{\prime}_{j}=S\cap\bigcup_{j=1}^{i}S_{j}$ and, therefore, $\bigcup_{i}S^{\prime}_{i}=S$. By the countable additivity of $\mu_{0}$,

 $\sum_{i}\mu_{0}(S_{i})=\sum_{i}(\mu_{0}(S^{\prime}_{i})+\mu_{0}(S_{i}\setminus S% ^{\prime}_{i}))\geq\sum_{i}\mu_{0}(S^{\prime}_{i})=\mu_{0}(S).$

As this inequality hold for any sequence $S_{i}\in A$ covering $S$, equation (1) gives $\mu^{*}(S)\geq\mu_{0}(S)$ and, by combining with the reverse inequality, shows that $\mu^{*}$ does indeed agree with $\mu_{0}$ on $A$.

We have shown that $\mu_{0}$ extends to an outer measure $\mu^{*}$ on the power set of $X$. The final step is to apply Carathéodory’s lemma on the restriction of outer measures. A set $S\subseteq X$ is said to be $\mu^{*}$-measurable if the inequality

 $\mu^{*}(E)\geq\mu^{*}(E\cap S)+\mu^{*}(E\cap S^{c})$ (3)

is satisfied for all subsets $E$ of $X$. Carathéodory’s lemma then states that the collection $\mathcal{F}$ of $\mu^{*}$-measurable sets is a $\sigma$-algebra (http://planetmath.org/SigmaAlgebra) and that the restriction of $\mu^{*}$ to $\mathcal{F}$ is a measure. To complete the proof of the theorem it only remains to be shown that every set in $A$ is $\mu^{*}$-measurable, as it will then follow that $\mathcal{F}$ contains $\mathcal{A}=\sigma(A)$ and the restriction of $\mu^{*}$ to $\mathcal{A}$ is a measure.

So, choosing any $S\in A$ and $E\subseteq X$, the proof will be complete once it is shown that (3) is satisfied. Given any $\epsilon>0$, equation (1) says that there is a sequence $E_{i}$ in $A$ such that $E\subseteq\bigcup_{i}E_{i}$ and

 $\sum_{i}\mu_{0}(E_{i})\leq\mu^{*}(E)+\epsilon.$

As $E\cap S\subseteq\bigcup_{i}(E_{i}\cap S)$ and $E\cap S^{c}\subseteq\bigcup_{i}(E_{i}\cap S^{c})$,

 $\mu^{*}(E\cap S)+\mu^{*}(E\cap S^{c})\leq\sum_{i}\mu_{0}(E_{i}\cap S)+\sum_{i}% \mu_{0}(E_{i}\cap S^{c})=\sum_{i}\mu_{0}(E_{i})\leq\mu^{*}(E)+\epsilon.$

Since $\epsilon$ is arbitrary, this shows that (3) is satisfied and $S$ is $\mu^{*}$-measurable.

Title proof of Carathéodory’s extension theorem ProofOfCaratheodorysExtensionTheorem 2013-03-22 18:33:28 2013-03-22 18:33:28 gel (22282) gel (22282) 4 gel (22282) Proof msc 28A12 CaratheodorysLemma Measure OuterMeasure2