TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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

1.1givenalgebra

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 $Y$ .

The trace of a sigma-algebra is a sigma-algebra on the traced subset

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources