`BookProof.ChapterAbelianCyclicModel.integral_repMeasure` (f : C(X, ℂ)) : inner ℂ xi (pi f xi) = ∫ x, f x ∂(repMeasure pi xi)

Proved

BookProof.ChapterAbelianCyclicModel.integral_repMeasure

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianCyclicModel.

BookProof.ChapterAbelianCyclicModel.integral_repMeasure (f : C(X, ℂ)) : inner ℂ xi (pi f xi) = ∫ x, f x ∂(repMeasure pi xi)

Formalization note: Lean 4 identifier BookProof.ChapterAbelianCyclicModel.integral_repMeasure.

Preamble

-- Generated from ChapterAbelianCyclicModel.lean — theorem BookProof.ChapterAbelianCyclicModel.integral_repMeasure
import Definitions.Def_ChapterAbelianGelfandModel
import Mathlib
import Definitions.Def_ChapterAbelianCyclicModel
open BookProof.ChapterAbelianCyclicModel


open MeasureTheory Complex WeakDual
open scoped ComplexOrder


open BookProof.ChapterAbelianGelfandModel


variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (pi : C(X, ℂ) →⋆ₐ[ℂ] (H →L[ℂ] H)) (xi : H)

Formal statement

theorem BookProof.ChapterAbelianCyclicModel.integral_repMeasure (f : C(X, ℂ)) :
    inner ℂ xi (pi f xi) = ∫ x, f x ∂(repMeasure pi xi) := by sorry

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