Measure, Integration & Real Analysis · Problem 12.7

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · independent-events-union

Problem

Suppose

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

is a probability space,

A \in \mathcal{F}

, and

B_1 \subseteq B_2 \subseteq \cdots

is an increasing sequence of events such that

A

and

B_n

are independent events for each

n \in \mathbf{Z}^+

. Show that

A

and

\bigcup_{n=1}^{\infty} B_n

are independent.