The class of $\sigma$-finite measures has some very convenient properties; $\sigma$-finiteness can be compared in this respect to separability of topological spaces. Some theorems in analysis require σ-finiteness as a hypothesis. For example, both the Radon–Nikodym theorem and Fubini's theorem are invalid without an assumption of $\sigma$-finiteness (or something similar) on the measures involved.
Though measures which are not σ-finite are sometimes regarded as pathological, ...
I was wondering what makes $\sigma$-finite measures so natural to mathematicians (they often think of them in the first place when it comes to measures, while I as a layman don't have that instinct), well-behaved (as opposite to "pathological") and important (appearing in conditions in many theorems such as Radon-Nikodym, Lebesgue decomposition and Fubini's Theorems)?
In what sense/respect, can $\sigma$-finiteness be compared to separability of topological spaces?
For example, are most or all properties true for finite measures also true for $\sigma$-finite measures, but not for general infinite measures? If yes, why is that?
Are all above because of equivalence of $\sigma$-finite measures to probability measures? If yes, how is it the reason?
Thanks and regards!