TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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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

1.1

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.

givenalgebra

2.1

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 $Y$ .

step 1.1givenalgebra∎

Depends on

The trace of a sigma-algebra on a subset

Used by

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 3 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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