Measure, Integration & Real Analysis · Problem 2D.9

Integration — Sheldon Axler

Mathematics — Measure Theory · lebesgue-measurable-dilation-invariant

Problem

Prove that the collection of Lebesgue measurable subsets of

\mathbf{R}

is dilation invariant. More precisely, prove that if

A \subseteq \mathbf{R}

is Lebesgue measurable and

t \in \mathbf{R}

, then

tA

(which is defined to be

\{ta : a \in A\}

) is Lebesgue measurable.