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 ${\mathcal {M}}$.

Proof

First, observe that ${\mathcal {M}}$ is closed under complements due to symmetry in the meaning of $\mu$-measurability. Now, we show if $A,B$ then $A\cup B\in {\mathcal {M}}$.