Measure, Integration & Real Analysis · Problem 12.9

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · conditional-probability-chain-rule

Problem

Suppose

(\Omega, \mathcal{F}, P)

is a probability space and

B_1, \ldots, B_n \in \mathcal{F}

are such that

P(B_1 \cap \cdots \cap B_n) > 0

. Prove that

P(A \cap B_1 \cap \cdots \cap B_n) = P(B_1) \cdot P_{B_1}(B_2) \cdots P_{B_1 \cap \cdots \cap B_{n-1}}(B_n) \cdot P_{B_1 \cap \cdots \cap B_n}(A)

for every event

A \in \mathcal{F}