Measure, Integration & Real Analysis · Problem 3A.1

Integration — Sheldon Axler

Mathematics — Measure Theory · integral-infinite-measure-set

Problem

Suppose (X,S,μ)(X, \mathcal{S}, \mu) is a measure space and f:X→[0,∞]f: X \to [0, \infty] is an S\mathcal{S}-measurable function such that ∫f dμ<∞\int f\, d\mu < \infty. Explain why inf⁡Ef=0\inf_E f = 0 for each set E∈SE \in \mathcal{S} with μ(E)=∞\mu(E) = \infty.

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