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?
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I would assume if the limit exists, then certainly. However, there may by some duality here with Infinite Disjunctions and Conjunctions which is riddled with nuance. – Graviton Oct 21 '22 at 03:15
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1Thanks @Graviton – Astral Oct 21 '22 at 03:44
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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.
Qiaochu Yuan
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What is the exact sequence you are referring to on the LHS, and why is it decreasing sequence? @Qiaochu Yuan – Astral Oct 21 '22 at 12:34
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@Astral: I mean the sequence of probabilities $p_n = \mathbb{P}(A_1 \cap \dots \cap A_n)$. You keep taking intersections so $A_1 \cap \dots \cap A_n$ is a decreasing sequence of sets, so the corresponding probabilities are also decreasing. – Qiaochu Yuan Oct 21 '22 at 16:10
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