DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

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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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 2 results over 2 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)