The question I'm trying to figure out is:
Let $m$ be a $\sigma$-additive measure defined on a semiring $\mathscr{S}_m$ and let $\mathscr{L}$ be the domain of the Lebesgue extension of $m$. Let $m'$ be $\sigma$-additive extension of m to a semiring $\mathscr{S}_m'$ such that \begin{align*} \mathscr{S}_m\subset \mathscr{S}_m'\subset\mathscr{L}, \end{align*}
and let $\mathscr{L'}$ be the domain of the Lebesgue extension of $m'$. Prove that $\mathscr{L'}$=$\mathscr{L}$.
I think that I need to look at the inner/ outer Lebesgue measure of each extension and the coverings of each semiring. But I'm not sure how to compare them and prove they are the same. Any help would be great!