Measure, Integration & Real Analysis · Problem 2C.10

Measures — Sheldon Axler

Mathematics — Measure Theory · measures

Problem

Give an example of a measure space

(X, \mathcal{S}, \mu)

and a decreasing sequence

E_1 \supseteq E_2 \supseteq \cdots

of sets in

\mathcal{S}

such that

\mu\!\left(\bigcap_{k=1}^{\infty} E_k\right) \neq \lim_{k \to \infty} \mu(E_k).