I have recently started studying the more measure theory side of probability, and a condition that comes up constantly is that a measure in question is $\sigma$-finite.
My textbook defines a $\sigma$-finite measure $\mu$ on $S$ as being a measure such that there exists sets $A_1, A_2, \dots$ where
$\cup_i A_i = S$
$\mu (A_i) < \infty $ for all $i$
I don't understand why this definition in particular is used so often. What is it about being able to find such a subsequence that is so important?
I'm hoping someone can either give me a simple intuitive explanation, or maybe an example of how them not being $\sigma$-finite can cause problems
It's the next best thing to having a space of finite measure.
It means you can build the space up out of chunks of finite measure.
It doesn't get you anything to have a finite union of such chunks (that's always true), so the next best thing is the "smallest" infinite union -- i.e., a countably infinite union.
For example, $\mathbb R$ doesn't have finite Lebesgue measure, but every interval $(a,b)$ does, and $\mathbb R$ is a countable union of such intervals.