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

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Normalized Haar measure on a finite group

Example

Assume AC. For a finite group $G$ of order $m$ , the normalized Haar probability is $μ (E) = ∣ E ∣ / m$ , and $\int f d μ = m^{- 1} \sum_{x \in G} f (x)$ . It is invariant on both sides and under inversion.

Facts & Assumptions

Given: A finite group of order $m \geq 1$ , with AC for the cited general uniqueness theorem.

[F1]

A compact Hausdorff group has a unique normalized Haar probability under AC. (Normalized Haar probability on a compact group)

[F2]

Counting measure on a discrete group is Haar and its integrals are finite sums. (Counting measure as Haar measure on a discrete group)

[A1]

AC is assumed in the choice-function form stated in the cited definition; the explicit finite formula itself uses no choice. (The Axiom of Choice)

Verification

technique · direct

1.1

With the discrete topology $G$ is compact and Hausdorff: its singleton sets are open, and finitely many of them cover $G$ . Counting measure has total mass $m > 0$ , so dividing by $m$ gives the stated Haar probability by [F2]. Left multiplication, right multiplication and inversion each permute the $m$ summands; hence their pullbacks preserve the displayed average. Under the assumed AC, [F1] identifies this probability with the general normalized Haar probability.

F1F2A1

2.1

For $G = C_{3} = {1, a, a^{2}}$ , take $f (1) = 0$ , $f (a) = 1$ , $f (a^{2}) = 5$ . Then $\int f d μ = (0 + 1 + 5) / 3 = 2$ . Left or right translation by $a$ and inversion only rearrange these three values, so their integrals remain $2$ . For $m = 1$ the formula is evaluation at the identity, and for $f = 0$ it is zero.

step 1.1∎

Sources

Knapp, Advanced Real Analysis, VI §2, pp.225–230, Lemmas 6.9–6.13. Local argument and conventions as displayed above.

Depends on

Used by

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Dependency tree · two levels

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Sources

Knapp, Advanced Real Analysis, VI §2, pp.225–230, Lemmas 6.9–6.13 (standard reference, not scraped)