Measure, Integration & Real Analysis · Problem 6B.8

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · integration-scalar-multiplication-complex

Problem

Suppose

(X, \mathcal{S}, \mu)

is a measure space and

f : X \to \mathbf{C}

is an

\mathcal{S}

-measurable function such that

\int |f| \, d\mu < \infty

. Prove that if

\alpha \in \mathbf{C}

, then

\int \alpha f \, d\mu = \alpha \int f \, d\mu.