DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Compact support, $C_{c} (X)$ , and $C_{0} (X)$

Definition

Let $X$ be locally compact Hausdorff and let $f : X \to F$ be continuous, where $F$ is $R$ or $C$ . Its support is $supp f : = \overline{{x : f (x) \neq 0}}$ . Put $C_{c} (X; F) = {f \in C (X; F) : supp f is compact}$ . Also $C_{0} (X; F)$ consists of those $f$ for which, for every $ϵ > 0$ , ${x : ∣ f (x) ∣ \geq ϵ}$ is compact. We write $C_{c} (X)$ for the real space until the bounded complex $C_{0}$ theorem is invoked; $C_{0} (X)$ later means the complex space when its scalar field matters.

Depends on

Used by

Positive linear functionals on C_c(X) Definition
The compactly supported cutoff relation f≺ U Definition
C_c(X) is dense in Lⁱnfinity(mu) for every Radon measure False statement
A bounded real C₀(X) functional is a difference of positive functionals Lemma
A finite compactly supported partition of unity near a compact set Lemma
LCH Urysohn cutoff Lemma
Positive C₀(X) functionals have finite regular representing measures Lemma
C_c(X) is dense in Lᵖ(mu) for a Radon measure Theorem
Sigma-compact open sets make locally finite Borel measures regular Theorem

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

Donald L. Cohn, Measure Theory, 2nd ed., §7.1 (standard reference, not scraped)

Compact support, Cc(X)Cc​(X), and C0(X)C0​(X)