DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-29

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The completed product measure

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(X, A, μ)$ and $(Y, B, ν)$ be sigma-finite measure spaces. The completed product measure is the completion of the measure $μ \times ν$ on $(X \times Y, A \otimes B)$ :

$\overline{μ \times ν} : = the completion of μ \times ν .$

Its sigma-algebra consists of sets that differ from an $(A \otimes B)$ -measurable set by a subset of a $(μ \times ν)$ -null set, exactly as in The completion domain and proposed completed set function of a measure space.

Depends on

Dependency tree · 2 levels

Sources

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.39