DefinitionDefinition: AI-adaptedProof: 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.

The semifinite part of a measure

Definition

Let $μ$ be a measure on $(X, A)$ . Its semifinite part is the set function

$μ_{s f} (E) : = \sup {μ (F) : F \in A, F \subseteq E, μ (F) < + \infty}, E \in A .$

The set inside the supremum is nonempty because it contains the value $μ (\emptyset) = 0$ , and its supremum exists in $[0, + \infty]$ by Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ . Assuming the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), Assuming countable choice, the semifinite part is a semifinite measure and equals the original measure exactly when it is semifinite ↗ proves that this set function is a semifinite measure and identifies when it equals $μ$ .

Depends on

Used by

Dependency tree · two levels

18 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 15 (standard reference, not scraped)