Measure, Integration & Real Analysis · Problem 5C.4

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · lebesgue-measure-translation-invariant

Problem

Show that Lebesgue measure on

\mathbf{R}^n

is translation invariant. More precisely, show that if

E \in \mathcal{B}_n

and

a \in \mathbf{R}^n

, then

a + E \in \mathcal{B}_n

and

\lambda_n(a + E) = \lambda_n(E)

, where

a + E = \{a + x : x \in E\}.