 Measurability criterion inside a finite measurable hull

For a regular outer measure and measurable $M$, the forward implication is countable additivity. For the reverse, choose a measurable hull of $A$ and intersect it with $M$ to obtain $A\subseteq B\subseteq M$ and $\mu (B)={\mu }^{\ast }(A)$. Measurability of $B$ splits $M\setminus A$ into $M\setminus B$ and $B\setminus A$, soThe assumed equality and finiteness force ${\mu }^{\ast }(B\setminus A)=0$. Every outer-measure-null set is measurable by Caratheodory measurability criterion, so $A=B\setminus (B\setminus A)$ is measurable. Finiteness prevents invalid subtraction of infinite masses.