Measure, Integration & Real Analysis · Problem 2B.14

Measures — Sheldon Axler

Mathematics — Measure Theory · measurable-function-limsup-liminf

Problem

Suppose (X,S)(X, \mathcal{S}) is a measurable space and f1,f2,…f_1, f_2, \ldots is a sequence of S\mathcal{S}-measurable functions from XX to [−∞,∞][-\infty, \infty]. Prove that lim sup⁡k→∞fk\limsup_{k \to \infty} f_k and lim inf⁡k→∞fk\liminf_{k \to \infty} f_k are S\mathcal{S}-measurable functions.

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