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

1.1F1

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 $I_{ϕ} (0) = 0$ .

2.1F1step 1.1∎

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.

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

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