`BookProof.ChapterAbelianGelfandModel.integral_stateMeasure_ofReal` (f : C(X, ℝ)) : psi (toC f) = ∫ x, (toC f) x ∂(stateMeasure psi hpos)

Proved

BookProof.ChapterAbelianGelfandModel.integral_stateMeasure_ofReal

by leonardopedro · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianGelfandModel.

BookProof.ChapterAbelianGelfandModel.integral_stateMeasure_ofReal (f : C(X, ℝ)) : psi (toC f) = ∫ x, (toC f) x ∂(stateMeasure psi hpos)

Formalization note: Lean 4 identifier BookProof.ChapterAbelianGelfandModel.integral_stateMeasure_ofReal.

Preamble

-- Generated from ChapterAbelianGelfandModel.lean — theorem BookProof.ChapterAbelianGelfandModel.integral_stateMeasure_ofReal
import Definitions.Def_ChapterLinftyMultiplication
import Mathlib
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterA3b
open BookProof.ChapterAbelianGelfandModel


open MeasureTheory Complex WeakDual CompactlySupported CompactlySupportedContinuousMap
open scoped ComplexOrder


open BookProof.ChapterLinftyMultiplication

variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
variable {X : Type*} [TopologicalSpace X]
variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
  (psi : C(X, ℂ) →ₗ[ℂ] ℂ) (hpos : ∀ g : C(X, ℂ), 0 ≤ psi (star g * g))

Formal statement

theorem BookProof.ChapterAbelianGelfandModel.integral_stateMeasure_ofReal (f : C(X, ℝ)) :
    psi (toC f) = ∫ x, (toC f) x ∂(stateMeasure psi hpos) := by sorry

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