`BookProof.ChapterAbelianGelfandModel.exists_probabilityMeasure_of_state_inc` (hone : psi 1 = 1) : ∃ mu : Measure X, IsProbabilityMeasure mu ∧ ∀ g : C(X, ℂ), psi g = ∫ x, g x ∂mu

Proved

BookProof.ChapterAbelianGelfandModel.exists_probabilityMeasure_of_state_inc

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianGelfandModel.

BookProof.ChapterAbelianGelfandModel.exists_probabilityMeasure_of_state_inc (hone : psi 1 = 1) : ∃ mu : Measure X, IsProbabilityMeasure mu ∧ ∀ g : C(X, ℂ), psi g = ∫ x, g x ∂mu

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

Replaces the retired node BookProof.ChapterAbelianGelfandModel.exists_probabilityMeasure_of_state, whose stored statement omitted the hpos hypothesis binder (dropped include).

Preamble

-- Generated from ChapterAbelianGelfandModel.lean — theorem BookProof.ChapterAbelianGelfandModel.exists_probabilityMeasure_of_state_inc
import Definitions.Def_ChapterLinftyMultiplication
import Mathlib
import Definitions.Def_ChapterAbelianGelfandModel
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]
  (psi : C(X, ℂ) →ₗ[ℂ] ℂ) (hpos : ∀ g : C(X, ℂ), 0 ≤ psi (star g * g))

include hpos in

Formal statement

theorem BookProof.ChapterAbelianGelfandModel.exists_probabilityMeasure_of_state_inc (hone : psi 1 = 1) :
    ∃ mu : Measure X, IsProbabilityMeasure mu ∧
      ∀ g : C(X, ℂ), psi g = ∫ x, g x ∂mu := by sorry

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