DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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 trace of a sigma-algebra on a subset

Definition

Let $A$ be a sigma-algebra on $X$ (Sigma-algebras) and let $Y \subseteq X$ . The trace of $A$ on $Y$ is the family

$A ∣_{Y} : = {A \cap Y : A \in A} \subseteq P (Y) .$

This definition forms a family of subsets of $Y$ . That it satisfies the sigma-algebra axioms on $Y$ is proved in The trace of a sigma-algebra is a sigma-algebra on the traced subset ↗.

Depends on

Sigma-algebras

Used by

First-return times and induced transformations Definition
First-return time and induced map are measurable Proposition
Generating a sigma-algebra commutes with taking traces Theorem
The trace of a sigma-algebra is a sigma-algebra on the traced subset Theorem

Dependency tree · two levels

2 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

T. Tao, An Introduction to Measure Theory, Exercise 1.4.12 (standard reference, not scraped)