If $m$ is an outer measure defined on a set $X$, we say that a subset $E$ of $X$ is Carathéodory-measurable with respect to $m$ if for all subsets $A$ of $X$, we have $m(A)=m(A\cap E) + m(A\cap E^c)$. And if $M$ is the set of all Carathéodory-measurable sets with respect to $m$, then $M$ is a complete sigma algebra on $X$ and $m$ restricted to $M$ is a complete measure on $X$.
My question is, is $M$ "optimal" in some way? Is it the biggest or smallest subset of $P(X)$ such that $m$ restricted to that subset is a measure? Is it the biggest or smallest subset of $P(X)$ such that $m$ restricted to that subset is a complete measure?
To put it another way, what is it that makes the Carathéodory measurability criterion the "best" criterion for measurability? Or is it not the best, but just an arbitrary choice out of a sea of infinitely many equally good stronger and weaker measurability criteria?
Edit: Yes, the Caratheodory condition defines the largest $\sigma$-algebra $\sum$ such that the restriction of $m$ to $\sum$ is a measure and such that we also have $$m(A)=\inf_{A\subset E\in\sum}m(E)$$for all $A\subset X$. That follows from this:
Proof. Suppose $A\subset X$ and $E\in\sum$. Choose $F$ with $A\subset F\in\sum$. Since $\mu$ is a measure, $A\cap E\subset F\cap E\in\sum$ and $A\setminus E\subset F\setminus E\in\sum$ we have $$\mu(F)=\mu(F\cap E)+\mu(F\setminus E)\ge\mu^*(A\cap E)+\mu^*(A\setminus E).$$Taking the inf over $F$ shows that $$\mu^*(A)\ge\mu^*(A\cap E)+\mu^*(A\setminus E).$$
The other inequality is clear, since it's easy to see that $\mu^*$ is subadditive: $\mu^*(B\cup C)\le\mu^*(B)+\mu^*(C)$ for every $B$ and $C$. (If $B\subset E_a\in\sum$ and $C\subset E_2\in\sum$ then $$\mu^*(B\cup C)\le\mu(E_a\cup E_2)\le\mu(E_1)+\mu(E_2);$$now take the inf over $E_1$ and $E_2$ on the right side.)