Carathéodory's Theorem (Measure Theory)

This proof is about Carathéodory's Theorem in the context of Measure Theory. For other uses, see Carathéodory's Theorem.

Theorem

Let $X$ be a set.

Let $\SS \subseteq \powerset X$ be a semi-ring of subsets of $X$.

Let $\mu: \SS \to \overline \R$ be a pre-measure on $\SS$.

Let $\map \sigma \SS$ be the $\sigma$-algebra generated by $\SS$.

Then $\mu$ extends to a measure $\mu^*$ on $\map \sigma \SS$.

Corollary

Suppose there exists an exhausting sequence $\sequence {S_n}_{n \mathop \in \N} \uparrow X$ in $\SS$ such that:

$\forall n \in \N: \map \mu {S_n} < +\infty$

Then the extension $\mu^*$ is unique.

Proof

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Source of Name

This entry was named for Constantin Carathéodory.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $6.1$