DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Haar covering ratio of test functions

Definition

Use $C_{c} (G)^{+} = {f \in C_{c} (G; R) : f \geq 0}$ and translations from Left Haar integral and left Haar measure, which preserve $C_{c}$ by Translations preserve compactly supported continuous functions. For $f \in C_{c} (G)^{+}$ and $0 \neq ϕ \in C_{c} (G)^{+}$ put $(f : ϕ) = \inf {\sum_{j = 1}^{n} c_{j} : f \leq \sum_{j = 1}^{n} c_{j} L_{x_{j}} ϕ, n \geq 0, c_{j} > 0, x_{j} \in G} .$ The empty sum is zero and $\inf \emptyset = + \infty$ ; thus $(0 : ϕ) = 0$ . Fix $0 \neq f_{0} \in C_{c} (G)^{+}$ . Once finiteness and strict positivity of $(f_{0} : ϕ)$ have been proved, write $I_{ϕ} (f) = (f : ϕ) / (f_{0} : ϕ)$ . The definition does not assume those properties before the following lemma proves them.

Sources

Pedersen, Haar integral, p.2 definitions and lemma; p.3 Theorem 1; pp.4–5 second proof and Remark 2. Local argument and conventions as displayed above.

Depends on

Dependency tree · 2 levels

Sources

Pedersen, Haar integral, p.2 definitions and lemma; p.3 Theorem 1; pp.4–5 second proof and Remark 2