Let's say we construct measure theory without the axiom of choice in our pocket. Since we don't have to worry about unmeasurable sets anymore (see this thread), is there any good reason one might define a measure on a $\sigma$-algebra other than the power set? If not, would $\sigma$-algebras still serve any purpose at all? Or are they used in analysis solely to sidestep problems of unmeasurability?
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The answer is a mixture of yes and no. $\sigma$-algebras serve the purpose of defining the collection of sets you want to measure. In measure theory with AC one may naively wish to measure all subsets (say of $\mathbb R$) but that is impossible and thus one must consider a $\sigma$-algebra strictly smaller than $\mathcal P (X)$. However, while it is true that in the absence of AC it may be the case that all subsets of $\mathbb R$ are measurable it still does not mean that you will necessarily want such a measure. For instance, if you are interested in Borel measurability then you don't really need the more general Lebesgue measure, let alone a measure in which all subsets are measurable.
So, much like anything at all, it all depends on what you want to do. Rarely are you interested in setting up a measure on as many sets as possible. Rather you want the measure to measure just what you need. Tailoring the measure to the purpose at hand is the point and if the most natural $\sigma$-algebra is strictly smaller than what is actually possible then so be it. This can be seen as a case of Polya's principle of the right amount of generality. There is no point of generalizing for the sake of generalization.
Non-measurability is not a problem per se, in other words, it is not a priori desirable to make every subset measurable, even when one can. In fact, conditioning (on non trivial sigma-algebras) is seen by some as the first and foremost distinctive tool used in probability. One wants to condition on sigma-algebras G which are neither the full power set nor the trivial sigma-algebra, just like, in $L^2$ spaces, one wants to consider projections which are neither the identity nor the mean. Then, subsets not in the sigma-algebra G are, even for a while, even in finite probability spaces, non-measurable.