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

Integration — Sheldon Axler

Mathematics — Measure Theory · lebesgue-measurable-sets-closed-approximation

Problem

Prove that if

A \subseteq \mathbf{R}

is Lebesgue measurable, then there exists an increasing sequence

F_1 \subseteq F_2 \subseteq \cdots

of closed sets contained in

A

such that

\left|A \setminus \bigcup_{k=1}^{\infty} F_k\right| = 0.