Measure, Integration & Real Analysis · Problem 3A.3

Integration — Sheldon Axler

Mathematics — Measure Theory · integral-positive-iff-positive-measure

Problem

Suppose

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

is a measure space and

f: X \to [0, \infty]

is an

\mathcal{S}

-measurable function. Prove that

\int f\, d\mu > 0 \text{ if and only if } \mu(\{x \in X : f(x) > 0\}) > 0.