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

Measures — Sheldon Axler

Mathematics — Measure Theory · measures

Problem

Suppose μ\mu is a measure on (Z+,2Z+)(\mathbf{Z}^+, 2^{\mathbf{Z}^+}). Prove that there is a sequence w1,w2,…w_1, w_2, \ldots in [0,∞][0, \infty] such that μ(E)=∑k∈Ewk\mu(E) = \sum_{k \in E} w_k for every set E⊆Z+E \subseteq \mathbf{Z}^+. [Let 2Z+2^{\mathbf{Z}^+} denote the σ\sigma-algebra on Z+\mathbf{Z}^+ consisting of all subsets of Z+\mathbf{Z}^+.]

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