Measure, Integration & Real Analysis · Problem 2C.9

Measures — Sheldon Axler

Mathematics — Measure Theory · measures

Problem

Suppose μ\mu and ν\nu are measures on a measurable space (X,S)(X, \mathcal{S}). Prove that μ+ν\mu + \nu is a measure on (X,S)(X, \mathcal{S}). [Here μ+ν\mu + \nu is the usual sum of two functions: if E∈SE \in \mathcal{S}, then (μ+ν)(E)=μ(E)+ν(E)(\mu + \nu)(E) = \mu(E) + \nu(E).]

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