Measure, Integration & Real Analysis · Problem 5A.8

Product Measures — Sheldon Axler

Mathematics — Measure Theory · sigma-finite-equivalences

Problem

Suppose μ\mu is a measure on a measurable space (X,S)(X, \mathcal{S}). Prove that the following are equivalent. (a) The measure μ\mu is σ\sigma-finite. (b) There exists an increasing sequence X1⊆X2⊆⋯X_1 \subseteq X_2 \subseteq \cdots of sets in S\mathcal{S} such that X=⋃k=1∞XkX = \bigcup_{k=1}^\infty X_k and μ(Xk)<∞\mu(X_k) < \infty for every k∈Z+k \in \mathbf{Z}^+. (c) There exists a disjoint sequence X1,X2,X3,…X_1, X_2, X_3, \ldots of sets in S\mathcal{S} such that X=⋃k=1∞XkX = \bigcup_{k=1}^\infty X_k and μ(Xk)<∞\mu(X_k) < \infty for every k∈Z+k \in \mathbf{Z}^+.

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