LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Normalized approximate Haar functionals are positive and invariant in the limit

Statement

Fix $0 \neq f_{0} \in C_{c} (G)^{+}$ . Every $I_{ϕ}$ , for $0 \neq ϕ \in C_{c} (G)^{+}$ , is nonnegative, strictly positive on nonzero arguments, positively homogeneous, exactly left invariant and satisfies $I_{ϕ} (f_{0}) = 1$ . It is subadditive and obeys the coordinate bounds $1 / (f_{0} : f) \leq I_{ϕ} (f) \leq (f : f_{0})$ for $f \neq 0$ , independently of $ϕ$ . Any coordinatewise limit obeys these same conditions, with value zero at $f = 0$ . Additivity is not asserted for the approximants.

Facts & Assumptions

Given: $f_{0}, ϕ$ nonzero and nonnegative; $f \geq 0$ .

[F1]

Ratios have finite positive denominators, homogeneity, invariance, subadditivity and the displayed coordinate bounds. (Haar covering ratios are finite and positive)

Proof

technique · direct

1.1

Dividing the numerator identities and inequalities by $(f_{0} : ϕ) > 0$ gives $I_{ϕ} (t f) = t I_{ϕ} (f)$ for $t \geq 0$ , $I_{ϕ} (L_{a} f) = I_{ϕ} (f)$ and subadditivity. The numerator at $f = f_{0}$ equals the denominator, hence $I_{ϕ} (f_{0}) = 1$ ; the zero numerator gives $I_{ϕ} (0) = 0$ .

2.1

For each nonzero $f$ the two finite bounds in [F1] are independent of $ϕ$ , and the lower bound is strictly positive. Each normalization, invariance or homogeneity equation involves finitely many coordinates and defines a closed subset of the real product; each inequality does too. Therefore every coordinatewise limit stays in these sets and retains the stated bounds and properties.

F1step 1.1∎

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

Haar covering ratios are finite and positive

Used by

Haar candidate sets have the finite intersection property Lemma

Dependency tree · two levels

4 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

Pedersen, Haar integral, p.2 definitions and lemma; p.3 Theorem 1; pp.4–5 second proof and Remark 2 (standard reference, not scraped)