Measure, Integration & Real Analysis · Problem 8C.2

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · unordered-sums-nonnegative

Problem

Suppose {ak}k∈Γ\{a_k\}_{k\in\Gamma} is a family in R\mathbf{R} and ak≥0a_k \geq 0 for each k∈Γk \in \Gamma. Prove the unordered sum ∑k∈Γak\sum_{k\in\Gamma} a_k converges if and only if sup⁡{∑j∈Ωaj:Ω is a finite subset of Γ}<∞.\sup\Bigl\{\sum_{j\in\Omega} a_j : \Omega \text{ is a finite subset of } \Gamma\Bigr\} < \infty. Furthermore, prove that if ∑k∈Γak\sum_{k\in\Gamma} a_k converges then it equals the supremum above.

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