We know that a $\sigma$-algebra is a collection of sets closed under countable set operations. My question is:
how was it determined that this is the right collection?
i.e., how was it determined that these are the sets that reasonably should be measurable.
As an analogue, we know that the definition of a topology comes from the behaviour of the collection of open sets in $\mathbb R^n$, and the fact that continuity may be rephrased as a statement about open sets. That is, the generalization of taking the essential property of behaviour of open sets and continuity is intuitive.
I've never heard nor read an explanation in the same spirit of this one for a $\sigma$-algebra. I'd assume in the same sense, this comes from what subsets of $\mathbb R^n$ we should expect to be Lebesgue Measurable, and then generalizing to arbitrary measures.
Or Is there some way to show this is the maximal collection of sets for any measure?
That is,
does the definition of a $\sigma$-algebra come from some maximality property, or does it come from some generalization of Lebesgue Measurable Sets?
I've done a bit of research on the topic and come up short of an answer, so any discussion of the subject would be appreciated.
The issue here is why require countable unions (certainly countable unions of closed sets are not closed) and the answer is related to the countable additivity of the Lebesgue measure, which would be harder to express if the family of sets considered were not a $\sigma$-algebra. Kolmogorov started the trend of assuming that the Lebesgue measure is countably additive, but notice that this is strictly speaking not required for many applications, and the obligatory nature of such an assumption has been questioned in the recent literature. Notice that in some models of the Zermelo-Fraenkel set theory, the Lebesgue measure is not $\sigma$-additive (look for "Feferman-Levy model" here and at MO).