I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined on a semiring $\mathscr{S}_m$ -and therefore uniquely extendible to the minimal ring $\mathscr{R}(\mathscr{S}_m)$- coincides with the sets of Lebesgue-measurable sets with respect to $m$. The statement, which is a proofless statement in the original text, appears in the form of problem 15 in an English translation here.
The fact that every Lebesgue-measurable set is a set of $\sigma$-uniqueness is virtually proven in the "hint", but what I find uncomprehensible is why the extension $\mu$ of $m$, which is defined on $\mathscr{R}(\mathscr{S}_m)$ and on $A$, is also defined on $A\triangle B$ with $B\in\mathscr{R}(\mathscr{S}_m)$ (we can find $\mu(A\triangle B)$ in the text): is that implicit in the definition of $\sigma$-additive extension of $m$ defined on $A$?
As to the converse, i.e. the fact that a $\sigma$-additive extension of $m$ to a larger system than the system of Lebesgue-measurable sets is not unique, I am not able to verify why it is true. Can anybody prove it or help me in any way?
I heartily thank you for any answer!!!