DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Restriction of a measure to a measurable set

Definition

Let $μ$ be a measure on $(X, A)$ and fix a measurable set $E \in A$ . The restriction of $μ$ to $E$ on the original sigma-algebra is the set function

$μ_{E} : A \to [0, + \infty], μ_{E} (A) : = μ (A \cap E) .$

This is the same-ambient convention: the domain remains $A$ , rather than becoming the trace sigma-algebra on $E$ . The fact that $μ_{E}$ is a measure is The restriction of a measure to a measurable set is a measure ↗.

Depends on

Measures on sigma-algebras

Used by

The restriction of a measure to a measurable set is a measure Proposition

Dependency tree · two levels

4 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

G. Folland, Real Analysis, 2nd ed., §1.3, Exercise 10 (standard reference, not scraped)