Prove2Me

`BookProof.ChapterG3.averagedMeasure_invariant` (G : Type*) [Group G] [Fintype G] {Z : Type*} [MeasurableSpace Z] [MulAction G Z] (hmeas : ∀ g : G, Measurable (fun x : Z => g • x))

Proved
BookProof.ChapterG3.averagedMeasure_invariant

by leonardopedro · Oct 18, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterG3.

BookProof.ChapterG3.averagedMeasure_invariant (G : Type*) [Group G] [Fintype G] {Z : Type*} [MeasurableSpace Z] [MulAction G Z] (hmeas : ∀ g : G, Measurable (fun x : Z => g • x)) (μ : Measure Z) (h : G) : (averagedMeasure G μ).map (fun x => h • x) = averagedMeasure G μ

Formalization note: Lean 4 identifier BookProof.ChapterG3.averagedMeasure_invariant.

Preamble
-- Generated from ChapterG3.lean — theorem BookProof.ChapterG3.averagedMeasure_invariant
import Mathlib
import Definitions.Def_ChapterG3
open BookProof.ChapterG3


open MeasureTheory
open scoped ENNReal



variable {X : Type*}
Formal statement
theorem BookProof.ChapterG3.averagedMeasure_invariant (G : Type*) [Group G] [Fintype G]
    {Z : Type*} [MeasurableSpace Z] [MulAction G Z]
    (hmeas : ∀ g : G, Measurable (fun x : Z => g • x))
    (μ : Measure Z) (h : G) :
    (averagedMeasure G μ).map (fun x => h • x) = averagedMeasure G μ := by sorry

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