Prove2Me

`BookProof.ChapterGaugeCasimirAverage.gaugeAverage_of_isPhysicalObservable` (μ : Measure G) [IsProbabilityMeasure μ] {f : X → ℝ} (hf : IsPhysicalObservable G f) : gaugeAverage (X :

Proved
BookProof.ChapterGaugeCasimirAverage.gaugeAverage_of_isPhysicalObservable

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeCasimirAverage.

BookProof.ChapterGaugeCasimirAverage.gaugeAverage_of_isPhysicalObservable (μ : Measure G) [IsProbabilityMeasure μ] {f : X → ℝ} (hf : IsPhysicalObservable G f) : gaugeAverage (X := X) μ f = f

Formalization note: Lean 4 identifier BookProof.ChapterGaugeCasimirAverage.gaugeAverage_of_isPhysicalObservable.

Preamble
-- Generated from ChapterGaugeCasimirAverage.lean — theorem BookProof.ChapterGaugeCasimirAverage.gaugeAverage_of_isPhysicalObservable
import Mathlib
import Definitions.Def_ChapterGaugeCasimirAverage
import Definitions.Def_ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeCasimirAverage



open BookProof.ChapterGaugeIncompleteFixing
open MeasureTheory
open scoped InnerProductSpace

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
variable {ι : Type*} [Fintype ι]
variable {X : Type*}
variable {G : Type*} [Group G] [MeasurableSpace G] [MeasurableMul G] [MulAction G X]
Formal statement
theorem BookProof.ChapterGaugeCasimirAverage.gaugeAverage_of_isPhysicalObservable (μ : Measure G) [IsProbabilityMeasure μ]
    {f : X → ℝ} (hf : IsPhysicalObservable G f) :
    gaugeAverage (X := X) μ f = f := by sorry

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