Caratheodory's Theorem

Statement

Consider an out measure $\mu$ on $X$. Define

${\mathcal {M}}=\{A\subseteq X:A\ {\text{is}}\ \mu -{\text{measurable}}\}$.

Then ${\mathcal {M}}$ is a $\sigma$-algebra and $\mu ^{*}$ is a measure on ${\mathcal {M}}$.