I'm trying to prove the following statement:
Let $(\Omega, P)$ be a probability space, and $A_1,A_2,\ldots$ be events with positive probabiliy. Show that: $P\left(\bigcap_{n=1}^{\infty}A_{n}\right)=P\left(A_{1}\right)\cdot\prod_{n=2}^{\infty}P\left(A_{n}\mid A_{1}\cap\ldots\cap A_{n-1}\right)$
I thought about proving the statement for $n\in \mathbb{N}$ using induction, and then take $n$ to infinity but it doesn't seem to work.
Any help would be appreciated.
Thank you!
Best Answer
HINT: let $B_m:=\bigcap_{n=1}^m A_n$, then $\{B_m\}_{m\in\mathbb{N}}$ is a decreasing sequence that converges (in the sense of the limits for sets) to $B_{\infty }:=\bigcap_{n=1}^{\infty }A_n$. Now use induction on $m$ and what you know about limits of probabilities on sequences of sets to prove your result.