When learning measure theory, for sure one encounters the Carathéodory extension theorem with extends a premeasure defined over a semiring to a measure defined over a $\sigma-$algebra. Let $X$ be a set, $\mathcal{R}$ be a semiring over $X$ and $\mu$ be a premeasure defined over $\mathcal{R}$.
For me, a natural question is: what if we apply the Carathéodory extension a second time to the measure space $(X,\mathcal{R}',\mu')$ itself obtained by the Carathéodory extension of $(X,\mathcal{R},\mu)$? Fortunately, the result stabilizes, because the outer measure induced by $\mu$ and $\mu'$ are the same: for any subset $S\subset X$ we have
$$\mu^*(S)\overset{\operatorname{def}}{=}\inf\left\{\sum^{\infty}_{n=1}\mu(A_n):A_n\in\mathcal{R},\bigcup^{\infty}_{n=1}A_n\supset S\right\}=\inf\{\mu'(A):A\in\mathcal{R}',A\supset S\},$$
where $\mu^*$ is the outer measure induced by $\mu$. Clearly we have "$\ge$" since $\displaystyle\bigcup^{\infty}_{n=1}A_n\in\mathcal{R}'$ and that $\displaystyle\sum^{\infty}_{n=1}\mu(A_n)\ge\mu'\left(\displaystyle\bigcup^{\infty}_{n=1}A_n\right)$. For "$\le$", note that $\mu'(A)=\mu^*(A)\ge\mu^*(S)$ for every $A\in\mathcal{R}'$.
So I would like to ask: is there a name designed for the measure spaces with the property "unable to be further extended by the Carathéodory process"? Are there any easy criteria to determine whether a measure space has such property?