Relation between Borel–Cantelli lemmas and Kolmogorov's zero-one law I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov's zero-one law?
The former is about limsup of a sequence of events, while the latter is about tail event of a sequence of independent sub sigma algebras or of independent random variables. Both have results for limsup/tail event to have either probability 0 or 1. I guess there are relations between but cannot identify them.
Can the former be viewed as a special case of the latter? How about in reverse direction?
Thanks and regards!

The first Borel-Cantelli is simply the fact that the probability of a union is at most the sum of the probabilities; it has nothing in common with Kolmogorov zero-one law.
What Kolmogorov zero-one law tells you in the setting of the second Borel-Cantelli lemma is that the probability of the limsup is $0$ or $1$, because (1) the limsup is always in the tail $\sigma$-algebra, (2) you are considering independent events, hence their tail $\sigma$-algebra is trivial. Then the second Borel-Cantelli lemma itself tells you that this probability is in fact $1$ under the non summability condition which you know.
