TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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 is a sigma-algebra on the traced subset

Statement

If $A$ is a sigma-algebra on $X$ and $Y \subseteq X$ , then the trace $A ∣_{Y}$ is a sigma-algebra on $Y$ .

Facts & Assumptions

Given: A sigma-algebra $A$ on $X$ , a subset $Y \subseteq X$ , and the trace $A ∣_{Y} = {A \cap Y : A \in A}$ of The trace of a sigma-algebra on a subset.

Proof

technique · direct

1.1givenalgebra

Since $\emptyset = \emptyset \cap Y$ , the empty set lies in the trace. If $A \cap Y$ lies in the trace, then $Y ∖ (A \cap Y) = (X ∖ A) \cap Y$ lies in it.

2.1step 1.1givenalgebra∎

For a sequence of traced sets, $⋃_{n} (A_{n} \cap Y) = (⋃_{n} A_{n}) \cap Y$ lies in the trace. Together with step 1.1 these are exactly the sigma-algebra axioms on $Y$ .

Depends on

The trace of a sigma-algebra on a subset

Used by

First-return times and induced transformations Definition
Generating a sigma-algebra commutes with taking traces Theorem

Cited to discharge well-definedness by The trace of a sigma-algebra on a subset.

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)