DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-27

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 sigma-algebra generated by a function

Definition

Let $f : X \to Y$ be a function and assume the target $Y$ carries a Borel sigma-algebra $B (Y)$ .

The sigma-algebra generated by $f$ is

$σ (f) : = {f^{- 1} (B) : B \in B (Y)} .$

Because preimages preserve complements and countable unions, this is a sigma-algebra on $X$ . It is the smallest sigma-algebra on $X$ that makes $f$ measurable.

The sigma-algebra generated by a function

Definition

Depends on

Dependency tree · 2 levels

Sources