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

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · lebesgue-measure-dilation-functions

Problem

Suppose

f : \mathbf{R}^n \to \mathbf{R}

\mathcal{B}_n

-measurable and

t \in \mathbf{R} \setminus \{0\}

. Define

f_t : \mathbf{R}^n \to \mathbf{R}

f_t(x) = f(tx)

. (a) Prove that

f_t

\mathcal{B}_n

-measurable. (b) Prove that if

\displaystyle\int_{\mathbf{R}^n} f \, d\lambda_n

is defined, then

\int_{\mathbf{R}^n} f_t \, d\lambda_n = \frac{1}{|t|^n} \int_{\mathbf{R}^n} f \, d\lambda_n.