Measure, Integration & Real Analysis · Problem 3A.10

Integration — Sheldon Axler

Mathematics — Measure Theory · integral-of-series

Problem

Suppose (X,S,μ)(X, \mathcal{S}, \mu) is a measure space and f1,f2,…f_1, f_2, \ldots is a sequence of nonneg\-ative S\mathcal{S}-measurable functions. Define f:X→[0,∞]f: X \to [0, \infty] by f(x)=∑k=1∞fk(x)f(x) = \sum_{k=1}^{\infty} f_k(x). Prove that ∫f dμ=∑k=1∞∫fk dμ.\int f\, d\mu = \sum_{k=1}^{\infty} \int f_k\, d\mu.

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