Measure, Integration & Real Analysis · Problem 12.8

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · independent-family-intersection

Problem

Suppose

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

is a probability space and

\{A_t\}_{t \in \mathbf{R}}

is an independent family of events such that

P(A_t) < 1

for each

t \in \mathbf{R}

. Prove that there exists a sequence

t_1, t_2, \ldots

\mathbf{R}

such that

P\bigl(\bigcap_{n=1}^{\infty} A_{t_n}\bigr) = 0