Divergence of conditional probabilities and Borel-Cantelli

Question

Let $(A_n)_{n \in \mathbb{N}}$ be a sequence of events on a probability space $(\Omega, \mathcal{F},P)$. I want to show that $P(\lim \sup A_n)=1$ iff $\sum_{n\in \mathbb{N}}P(A_n |A)$ diverges for every $A$ such that $P(A)>0$. The "if" part follows from the first Borel-Cantelli Lemma, but I have no clue about the converse part.

Answer 1

• If $\mathbb P\left(\limsup_nA_n \right)=1$, then the random variable $\sum_n\mathbb 1_{A_n}$ is almost surely infinite and if $A$ has a positive measure, then $\mathbb 1_A \sum_n\mathbb 1_{A_n}$ is not integrable (it is infinite on a set of positive measure) hence the series $\sum_n\mathbb P\left(A_n\mid A\right)$ diverges.

• Suppose that $\mathbb P\left(\limsup_nA_n \right)\lt 1$. Let $A:=\liminf_n A_n^c$. Then $A\cap A_n=\bigcup_{i\geqslant 1}\bigcap_{j\geqslant i}A_j^c\cap A_n= A_n\cap\bigcup_{j\geqslant n+1}A_j^c$. These sets are pairwise disjoint hence $\sum_n\mathbb P\left(A\cap A_n\right)$ converges.