Measure, Integration & Real Analysis · Problem 9A.5

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · integration-change-of-measure

Problem

Suppose

\mu

is a (positive) measure on a measurable space

(X, \mathcal{S})

and

h

is a nonnegative function in

\mathcal{L}^1(\mu)

. Let

\nu

be the (positive) measure on

(X, \mathcal{S})

defined by

d\nu = h\, d\mu

. Prove that

\int f\, d\nu = \int f h\, d\mu

for all

\mathcal{S}

-measurable functions

f: X \to [0, \infty]