DefinitionDefinition: AI-adaptedProof: Not applicablejudge 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 sigma-algebra and its finite iterates

Let $(X, A)$ and $(Y, B)$ be measurable spaces. Their product sigma-algebra is

$A \otimes B : = σ ({A \times B : A \in A, B \in B}),$

the sigma-algebra on $X \times Y$ generated by the measurable rectangles of Measurable rectangles in a product of measurable spaces.

For the empty family, put $\prod_{i < 0} X_{i} : = {()}$ and define its empty product sigma-algebra by

$⨂_{i < 0} A_{i} : = {\emptyset, {()}} .$

For a nonempty finite family of measurable spaces $(X_{i}, A_{i})$ with $i < n$ , the finite product sigma-algebra is defined recursively by

$⨂_{i < 1} A_{i} : = A_{0}, ⨂_{i < n + 1} A_{i} : = (⨂_{i < n} A_{i}) \otimes A_{n} .$

This recursive definition is the finite base case used later when the library passes to countable product constructions.

Depends on