DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

Means on bounded functions on a group

Definition

Let $G$ be a group. Write $ℓ^{\infty} (G)$ for the real vector space of bounded functions $f : G \to R$ .

A mean on $ℓ^{\infty} (G)$ is a linear map

$m : ℓ^{\infty} (G) \to R$

such that:

$f \geq 0$ pointwise implies $m (f) \geq 0$ ;
$m (1_{G}) = 1$ , where $1_{G}$ is the constant function $1$ .

Positivity and normalization imply $∣ m (f) ∣ \leq ∥ f ∥_{\infty}$ , so a mean is automatically bounded of norm $1$ .

Depends on

Group and abelian group

Used by

Left translation on bounded functions Definition
Left-invariant means and amenable groups Definition

Dependency tree · two levels

6 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

Cornelia Drutu and Michael Kapovich, Lectures on Geometric Group Theory (standard reference, not scraped)
C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)