Measure, Integration & Real Analysis · Problem 2D.6

Integration — Sheldon Axler

Mathematics — Measure Theory · lebesgue-measurability-finite-measure

Problem

Suppose

A \subseteq \mathbf{R}

and

|A| < \infty

. Prove that

A

is Lebesgue measurable if and only if for every

\varepsilon > 0

there exists a set

G

that is the union of finitely many disjoint bounded open intervals such that

|A \setminus G| + |G \setminus A| < \varepsilon