Measure, Integration & Real Analysis · Problem 3A.11

Integration — Sheldon Axler

Mathematics — Measure Theory · summable-integrals-pointwise-zero

Problem

Suppose (X,S,μ)(X, \mathcal{S}, \mu) is a measure space and f1,f2,…f_1, f_2, \ldots are S\mathcal{S}-measurable functions from XX to R\mathbf{R} such that ∑k=1∞∫∣fk∣ dμ<∞\sum_{k=1}^{\infty} \int |f_k|\, d\mu < \infty. Prove that there exists E∈SE \in \mathcal{S} such that μ(X∖E)=0\mu(X \setminus E) = 0 and lim⁡k→∞fk(x)=0\lim_{k \to \infty} f_k(x) = 0 for every x∈Ex \in E.

Browse the problem library