Let $\mu$ be a signed measure, define $$ |\mu|(E) = \mu^+(E) +\mu^-(E) $$ Why is it that $$ |{\mu}|(E) = \text{sup}\sum_{k=1}^n|\mu(E_k)|, $$ where the supremum is taken over all finite disjoint families $\{E_k\}_{k=1}^n$ of measurable subsets of $X$.
2026-03-25 07:38:45.1774424325
Signed Measure equivalent definition
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