Measure, Integration & Real Analysis · Problem 3A.9

Integration — Sheldon Axler

Mathematics — Measure Theory · integral-defines-measure

Problem

Suppose μ\mu is a measure on a measurable space (X,S)(X, \mathcal{S}) and f:X→[0,∞]f: X \to [0, \infty] is an S\mathcal{S}-measurable function. Define ν:S→[0,∞]\nu: \mathcal{S} \to [0, \infty] by ν(A)=∫χAf dμ\nu(A) = \int \chi_A f\, d\mu for A∈SA \in \mathcal{S}. Prove that ν\nu is a measure on (X,S)(X, \mathcal{S}).

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