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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

  1. 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)?
  2. 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?
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  • $\begingroup$ Well BCL2 works with pairwise independence. Is there an analogue for pairwise independence? Also I have a feeling the answer may involve the term 'sigma- finite' $\endgroup$
    – BCLC
    Jan 2, 2017 at 15:06
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    $\begingroup$ @BCLC I suspect you're right that an assumption like sigma-finiteness needs to be made about the measure, but I haven't worked it out. It's not clear to me if the analogue of pairwise independence would be any more or less natural than independence outside probability theory. $\endgroup$ Jan 3, 2017 at 1:23
  • $\begingroup$ There is no notion of independence for infinite measures. If you define independence in the usual way we cannot even say that independence of A and B implies that of $A^{c}$ and B, for example. BCL1 is quite often used for infinite measures also. $\endgroup$ Apr 12, 2018 at 6:13
  • $\begingroup$ Considered bounty (I mean, you do have 23,000 rep soooo....)? $\endgroup$
    – BCLC
    Apr 15, 2018 at 17:01
  • $\begingroup$ Dumb question: Did you know Any σ-finite measure μ on a space X is equivalent to a probability measure on X ? $\endgroup$
    – BCLC
    Apr 15, 2018 at 17:02

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