The first Borel-Cantelli Lemma says that if we have events $E_i$ and $\sum_iP(E_i)<\infty$ then the probability infinitely many events occur is 0. The second is a partial converse saying if the events are independent and $\sum_i P(E_i) = \infty$ than the probability infinitely many events occur is 1.
It's clear that the first has an analogue on non-probability measure spaces, namely that for a sequence of measurable sets $A_i$, if $\sum_i \mu(A_i) < \infty$ then $\mu(\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k) = 0.$ However, because of the independence assumption, the analogue of the second seems much less natural since I've never seen independence formulated outside probability theory. On the other hand, I've never really studied analysis too deeply.
My question is
- Is the analogue of the first Borel-Cantelli lemma used frequently in non-probabilistic contexts? Is there another nice interpretation of it that has nothing to do with probability (other than just a mechanical interpretation of the proof)?
- Does the purely measure-theoretical analogue of independence ever come up naturally outside probability theory? If so, does the second Borel-Cantelli lemma have utility outside probability theory?