Let $\{A_n\}$ be a sequence of events which may NOT be independent. We are asked to prove the following:
$$ P(A_n\ \text{i.o.}) = 1 \iff \text{for all $A$ with $P(A)>0$,}\ \sum_{n=1}^\infty P(A\cap A_n) = \infty. $$
Here is what I have so far. I think if we assume that ∑ P(A ∩ An) = ∞, That means P[(A ∩ An) i.o.]=1, that is the lim sup (A ∩ An) =1. If I am not wrong,lim sup (A ∩ An) is a subset of lim sup (An).Therefore P(lim sup (An)) must be greater than or equal to P (lim sup (A ∩ An)) and we have proved that P[An i.o.]= P( lim sup An ) = 1.
If we assume P[An i.o.]= P( lim sup An ) = 1, how can we prove this implies ∑ P(A ∩ An) = ∞ ? Once again please remember we cannot use independence.
can someone enlighten me on this one please :)