Measure, Integration & Real Analysis · Problem 2D.7

Integration — Sheldon Axler

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

Problem

Prove that if

A \subseteq \mathbf{R}

is Lebesgue measurable, then there exists a decreasing sequence

G_1 \supseteq G_2 \supseteq \cdots

of open sets containing

A

such that

\left|\left(\bigcap_{k=1}^{\infty} G_k\right) \setminus A\right| = 0.