Measure, Integration & Real Analysis · Problem 12.4

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · independent-family-complement

Problem

Suppose (Ω,F,P)(\Omega, \mathcal{F}, P) is a probability space and {Ak}k∈Γ\{A_k\}_{k \in \Gamma} is a family of events. Prove the family {Ak}k∈Γ\{A_k\}_{k \in \Gamma} is independent if and only if the family {Ω∖Ak}k∈Γ\{\Omega \setminus A_k\}_{k \in \Gamma} is independent.

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