= Solution
A <regular outer measure> has, for every $E\subseteq X$, a measurable hull $H\supseteq E$ with $\mu(H)=\mu(E)$.
Suppose $\mathcal C$ is an algebra containing $\mathfrak M_\mu^*$ and $\mu$ is finitely additive on $\mathcal C$. Fix $A\in\mathcal C$ and an arbitrary $E$ of finite <outer measure>. Take a measurable hull $H$ of $E$. Both $H\cap A$ and $H\setminus A$ belong to $\mathcal C$, so
$$
\mu(E)=\mu(H)=\mu(H\cap A)+\mu(H\setminus A)\geq\mu(E\cap A)+\mu(E\setminus A).
$$
Outer subadditivity gives equality. If $\mu(E)=\infty$, outer subadditivity already forces the sum on the right to be infinite, again giving equality. Therefore $A$ satisfies the <Caratheodory criterion>. We have proved $\mathcal C\subseteq\mathfrak M_\mu^*$, so
$$
\boxed{\mathcal C=\mathfrak M_\mu^*.}
$$
This proves the <maximal additivity algebra of a regular outer measure> without any sigma-finiteness assumption.
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