DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The product measure of two sigma-finite measure spaces

Definition

Let $(X, A, μ)$ and $(Y, B, ν)$ be sigma-finite measure spaces. For $E \in A \otimes B$ , define

$(μ \times ν) (E) : = \int_{X} ν (E_{x}) d μ = \int_{Y} μ (E^{y}) d ν .$

The two displayed integrals are well-defined by For sigma-finite measures, the section-measure functions are measurable and equal by For sigma-finite measures, the two section-measure integrals of a measurable set agree. This is the product measure of $μ$ and $ν$ .

Depends on

Used by

Dependency tree · two levels

11 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

Terence Tao, An Introduction to Measure Theory, Section 1.7.3 (standard reference, not scraped)