= 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 $\mu^*(E)\geq\mu^*(E\cap A)+\mu^*(E\setminus A)$. 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.
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