DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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A complex measure is a finite-valued countably additive set function

Definition

Let $(X, A)$ be a measurable space. A complex measure on $(X, A)$ is a function $ν : A ⟶ C$ such that:

$ν (\emptyset) = 0$ ;
for every pairwise disjoint sequence $(E_{n})_{n \in N}$ in $A$ , $ν (⋃_{n \in N} E_{n}) = \sum_{n = 0}^{\infty} ν (E_{n})$ in $C$ .

The codomain is the field $C$ ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ), so a complex measure is finite-valued by definition: there is no complex number called $+ \infty$ or $- \infty$ to allow.

Depends on

Used by

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Sources

John K. Hunter, Measure Theory, Definition 6.29 (standard reference, not scraped)
Richard F. Bass, Real Analysis for Graduate Students, Exercise 13.3 (standard reference, not scraped)