`BookProof.ChapterG2.integral_haarAverage` [MeasurableSpace X] [MeasurableSMul G X] [MeasurableMul G] (μ : Measure X) [IsProbabilityMeasure μ] (hμ : ∀ g : G, MeasurePreserving (fun

Proved

BookProof.ChapterG2.integral_haarAverage

by leonardopedro · Oct 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterG2.

BookProof.ChapterG2.integral_haarAverage [MeasurableSpace X] [MeasurableSMul G X] [MeasurableMul G] (μ : Measure X) [IsProbabilityMeasure μ] (hμ : ∀ g : G, MeasurePreserving (fun x : X => g • x) μ μ) (f : X → ℝ) (hint : Integrable (fun p : G × X => f (p.1⁻¹ • p.2)) (μG.prod μ)) : ∫ x, haarAverage (μG

Formalization note: Lean 4 identifier BookProof.ChapterG2.integral_haarAverage.

Preamble

-- Generated from ChapterG2.lean — theorem BookProof.ChapterG2.integral_haarAverage
import Mathlib
import Definitions.Def_ChapterG2
import Definitions.Def_ChapterG
open BookProof.ChapterG2
open BookProof.ChapterG



open MeasureTheory ProbabilityTheory
open scoped ProbabilityTheory

variable {Ω : Type*} [MeasurableSpace Ω]
variable {A : Type*} [CommRing A] (Q : A)
variable {G : Type*} [Group G] [MeasurableSpace G]
variable {μG : Measure G} [IsProbabilityMeasure μG] [μG.IsMulLeftInvariant]
variable {X : Type*} [MulAction G X]

Formal statement

theorem BookProof.ChapterG2.integral_haarAverage [MeasurableSpace X] [MeasurableSMul G X]
    [MeasurableMul G]
    (μ : Measure X) [IsProbabilityMeasure μ]
    (hμ : ∀ g : G, MeasurePreserving (fun x : X => g • x) μ μ)
    (f : X → ℝ)
    (hint : Integrable (fun p : G × X => f (p.1⁻¹ • p.2)) (μG.prod μ)) :
    ∫ x, haarAverage (μG := μG) f x ∂μ = ∫ x, f x ∂μ := by sorry

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