Maximal additivity algebra of a regular outer measure

For a regular outer measure, its Caratheodory measurability criterion defines a maximal algebra of sets on which that outer measure is finitely additive. Indeed, suppose an additive algebra contains all measurable sets and also $A$. Given $E$ of finite outer measure, take a measurable hull $H\supseteq E$ of equal mass. Finite additivity splits $H$ by $A$, and monotonicity gives ${\mu }^{\ast }(E)\ge {\mu }^{\ast }(E\cap A)+{\mu }^{\ast }(E\setminus A)$. Outer subadditivity gives equality. For infinite mass that equality already follows from outer subadditivity. Thus $A$ was measurable. No sigma-finite measure assumption is necessary.