Measure, Integration & Real Analysis · Problem 12.10

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · total-probability-formula

Problem

Suppose

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

is a probability space and

A \in \mathcal{F}

is an event such that

0 < P(A) < 1

. Prove that

P(B) = P_A(B) \cdot P(A) + P_{\Omega \setminus A}(B) \cdot P(\Omega \setminus A)

for every event

B \in \mathcal{F}