Sigma-Algebras and Measures - Applications

TheoremCaratheodory's Extension Theorem

Let $\mathcal{A}$ be an algebra of subsets of $X$ , and let $\mu_0: \mathcal{A} \to [0, \infty]$ be a measure on $\mathcal{A}$ . Assume that $\mu_0$ is $\sigma$ -finite on $\mathcal{A}$ , meaning there exists a sequence $\{A_n\} \subseteq \mathcal{A}$ with $X = \bigcup_{n=1}^{\infty} A_n$ and $\mu_0(A_n) < \infty$ for all $n$ .

Then $\mu_0$ can be extended to a measure $\mu$ on $\sigma(\mathcal{A})$ , the sigma-algebra generated by $\mathcal{A}$ . Moreover, this extension is unique.

This theorem is fundamental for constructing measures. It shows that to define a measure on a sigma-algebra, it suffices to define it on a generating algebra, as long as the pre-measure satisfies countable additivity on the algebra.

ExampleConstruction of Lebesgue Measure

To construct Lebesgue measure on $\mathbb{R}$ , we start with the algebra $\mathcal{A}$ of finite unions of intervals. Define $\mu_0$ on intervals by their length: $\mu_0\left((a, b]\right) = b - a$

Extend this additively to finite unions. Caratheodory's Extension Theorem guarantees that $\mu_0$ extends uniquely to the Borel sigma-algebra $\mathcal{B}(\mathbb{R})$ , giving Lebesgue measure on Borel sets.

Remark

The $\sigma$ -finiteness condition is essential for uniqueness. Without it, multiple extensions may exist. For example, on an uncountable set with the power set sigma-algebra, counting measure restricted to finite sets can be extended to the full power set in multiple ways.

The construction process involves first defining an outer measure $\mu^*$ on all subsets of $X$ : $\mu^*(E) = \inf \left\{\sum_{i=1}^{\infty} \mu_0(A_i) : E \subseteq \bigcup_{i=1}^{\infty} A_i, A_i \in \mathcal{A}\right\}$

Then one shows that sets in $\sigma(\mathcal{A})$ are $\mu^*$ -measurable according to the Caratheodory criterion, and that $\mu^*$ restricted to $\sigma(\mathcal{A})$ is the desired extension. This systematic approach makes Caratheodory's theorem the standard method for constructing measures in practice.