Is the multiplicative (chain) rule for conditional probability valid for infinitely many events?

Question

So for $n$ events $A_{1}, ...,A_{n}$ the chain rule for conditional probability states that

$P(A_{1} \cap....\cap A_{n}) = P(A_{1})\times... \times P(A_{n}|A_{1}\cap....\cap A_{n-1})$

What about infinitely many events?

Answer 1

Yes, there's no issue. The LHS is a (weakly) decreasing sequence of non-negative real numbers so it must have a limit, and same for the RHS, and those limits must agree because they are the same sequence.