In my cource of measure theory, Lebesgue measure were built starting with measure on semiring $S\subset 2^X$, after we defined outer measure for each $A\in2^X$ by $\inf\{ \sum{m[B] : A\subset \cup B , B \subset S} \}$, defined what are Lesbegue measurable sets, and prooved that they form $\sigma$-algebra. What confuse me here - is why we start this construction with semiring. Because:
As i undestood from here and here, what we realy need to define $\sigma$-algebra of Lebesgue measurable sets - is countably subadditive outer measure. Proof, that $\inf\{ \sum{m[B] : A\subset \cup B , B \subset S} \}$ is countably subadditive - uses only that $S$ is cover of $X$ (proof given with geometric progression)
In wikipedia Lebesgue measure for $ℝ$ constructed from topology of $ℝ$, and only common thing between topology and semiring with $E$ (biggets set), is that both of them are covers, again.
In this answer pointed that any system of sets could be used to generate minimal $\sigma$-algebra.
So, my question is - can we use any cover to generate Lebesgue measure on measurable sets of $2^X$, and Lebesgue measure generated by topology/semiring - are just cases of it? If not, how do we generalize semiring and topology in this perspective?
Finally, i understand all theorems about semirings in my cource, but still this structure seems to me very strange, i just dont feel it. I would appreciate if you would explain me intuition behind it, may be formulate it in more abstract terms. My guess - outer measure, generated by semiring, fulfills some criterias, but which ones?