Measure, Integration & Real Analysis · Problem 8C.6

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · unordered-sums-absolute-convergence-real

Problem

Suppose

\{a_k\}_{k\in\Gamma}

is a family in

\mathbf{R}

. Prove that the unordered sum

\sum_{k\in\Gamma} a_k

converges if and only if

\sum_{k\in\Gamma} |a_k| < \infty