Carathéodory's Theorem (Measure Theory)

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This proof is about Carathéodory's Theorem in measure theory. For other uses, see Carathéodory's Theorem.

Theorem

Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a semi-ring of subsets of $X$.

Let $\mu: \mathcal S \to \overline{\R}$ be a pre-measure on $\mathcal S$.

Let $\sigma \left({\mathcal S}\right)$ be the $\sigma$-algebra generated by $\mathcal S$.


Then $\mu$ extends to a measure $\mu^*$ on $\sigma \left({\mathcal S}\right)$.


Corollary

Suppose there exists an exhausting sequence $\left({S_n}\right)_{n \in \N} \uparrow X$ in $\mathcal S$ such that:

$\forall n \in \N: \mu \left({S_n}\right) < +\infty$

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


Proof



Source of Name

This entry was named for Constantin Carathéodory.


Sources