LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-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.

A positive linear functional on $C_{c} (X)$ is monotone

Statement

If $f, g \in C_{c} (X)$ and $f \leq g$ , then $Λ (f) \leq Λ (g)$ .

Facts & Assumptions

Given: A positive real-linear $Λ : C_{c} (X) \to R$ and $f \leq g$ .

Proof

technique · direct

1.1

Since supports of $f$ and $g$ are compact, $g - f \in C_{c} (X)$ ; moreover $g - f \geq 0$ .

given

2.1

Positivity gives $0 \leq Λ (g - f) = Λ (g) - Λ (f)$ , which is the claim.

step 1.1algebra∎

Depends on

Positive linear functionals on $C_c(X)$

Used by

The RMK functional outer content is well defined Lemma

Dependency tree · two levels

2 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., §7.2 (standard reference, not scraped)

A positive linear functional on Cc(X)Cc​(X) is monotone