Measure, Integration & Real Analysis · Problem 2B.20

Measures — Sheldon Axler

Mathematics — Measure Theory · measurable-functions

Problem

Suppose

(X, \mathcal{S})

is a measurable space and

f, g : X \to \mathbf{R}

are

\mathcal{S}

-measurable functions. Prove that if

f(x) > 0

for all

x \in X

, then

f^g

(which is the function whose value at

x \in X

equals

f(x)^{g(x)}

) is an

\mathcal{S}

-measurable function.