# Probability of a countable intersection

It's well know and easy to prove that given a finite number of events $$A_1,\dots,A_n$$ we can factorize the probability of their intersection as: $$\mathbb{P}(A_1\cap\dots\cap A_n)=\mathbb{P}(A_1)\mathbb{P}(A_2|A_1)\dots\mathbb{P}(A_n|A_1\cap \dots \cap A_n)$$ I was wondering if this remains true for a sequence of events, i.e. can we factorize a countable intersection in the same way?

Yes, just take limit on both sides to get $$P(\cap_n A_n)=P(A_1)\prod_{k=2}^{\infty} P(A_k|A_1,A_2,...,A_{k-1})$$.
• Right, the events $A_1\cap A_2\cap...\cap A_n$ decrease to the event $\cap_n A_n$ so we can pass to the limit. – Kabo Murphy Jun 7 at 11:58