Measure, Integration & Real Analysis · Problem 6B.6

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · measurability-algebraic-combinations-complex

Problem

Suppose (X,S)(X, \mathcal{S}) is a measurable space and f,g:X→Cf, g : X \to \mathbf{C} are S\mathcal{S}-measurable. Prove that (a) f+gf + g, f−gf - g, and fgfg are S\mathcal{S}-measurable functions; (b) if g(x)≠0g(x) \neq 0 for all x∈Xx \in X, then fg\dfrac{f}{g} is an S\mathcal{S}-measurable function.

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