Measure, Integration & Real Analysis · Problem 2A.5

Measures — Sheldon Axler

Mathematics — Measure Theory · closed-sets-finite-intersection-property

Problem

Suppose

\mathcal{A}

is a set of closed subsets of

\mathbf{R}

such that

\bigcap_{F \in \mathcal{A}} F = \varnothing

. Prove that if

\mathcal{A}

contains at least one bounded set, then there exist

n \in \mathbf{Z}^+

and

F_1, \ldots, F_n \in \mathcal{A}

such that

F_1 \cap \cdots \cap F_n = \varnothing