# Carathéodory's Theorem (Measure Theory)

*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

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