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The completed product measure
Definition
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be sigma-finite measure spaces. The completed product measure is the completion of the measure on :
Its sigma-algebra consists of sets that differ from an -measurable set by a subset of a -null set, exactly as in The completion domain and proposed completed set function of a measure space.
Depends on
Used by
- FALSE: every section of a completed-product measurable function is measurable False statement
- FALSE: the product of two complete measure spaces is complete False statement
- The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures Theorem
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability Theorem
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.39 (standard reference, not scraped)