Probability of a countable intersection

Question

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?

Answer 1

Yes, just take limit on both sides to get $P(\cap_n A_n)=P(A_1)\prod_{k=2}^{\infty} P(A_k|A_1,A_2,...,A_{k-1})$.