Let $\mathcal{A}$ be an algebra over a set $E$, and $\mu$ be a pre-measure over $\mathcal{A}$. We can define an outer measure $\mu^*$ over $2^E$, and a $\sigma-$algebra $\mathcal{M}$ over $E$ (by Caratheodory theorem). In fact $\mu^*\mid_{\mathcal{M}}$ is a measure $\mu$. (cf. Real Analysis by Folland P29, 30, 31).
I'm thus wondering whether there is a characterization of $\mathcal{M}$ and $m$. That's to say, is $\mathcal{M}$ the maximal $\sigma-$algebra s.t. there exists a measure $k$ with $k\mid_{\mathcal{A}} = \mu$ with [some property]?
