`BookProof.ChapterMeasureAtomicDiffuse.restrict_atomSet_eq_sum_dirac` [IsFiniteMeasure mu] : mu.restrict (atomSet mu) = Measure.sum (fun x : atomSet mu => mu {(x : α)} • Measure.di

Proved

BookProof.ChapterMeasureAtomicDiffuse.restrict_atomSet_eq_sum_dirac

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMeasureAtomicDiffuse.

BookProof.ChapterMeasureAtomicDiffuse.restrict_atomSet_eq_sum_dirac [IsFiniteMeasure mu] : mu.restrict (atomSet mu) = Measure.sum (fun x : atomSet mu => mu {(x : α)} • Measure.dirac (x : α))

Formalization note: Lean 4 identifier BookProof.ChapterMeasureAtomicDiffuse.restrict_atomSet_eq_sum_dirac.

Preamble

-- Generated from ChapterMeasureAtomicDiffuse.lean — theorem BookProof.ChapterMeasureAtomicDiffuse.restrict_atomSet_eq_sum_dirac
import Mathlib
import Definitions.Def_ChapterMeasureAtomicDiffuse
open BookProof.ChapterMeasureAtomicDiffuse


noncomputable section

open MeasureTheory Complex



variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α] (mu : Measure α)

Formal statement

theorem BookProof.ChapterMeasureAtomicDiffuse.restrict_atomSet_eq_sum_dirac [IsFiniteMeasure mu] :
    mu.restrict (atomSet mu)
      = Measure.sum (fun x : atomSet mu => mu {(x : α)} • Measure.dirac (x : α)) := by sorry

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