`BookProof.ChapterAtomicDecomposition.atomicPart_eq_sum_dirac` (mu : Measure X) [SFinite mu] : atomicPart mu = Measure.sum (fun x : atoms mu => mu {(x : X)} • Measure.dirac (x : X)

Proved

BookProof.ChapterAtomicDecomposition.atomicPart_eq_sum_dirac

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAtomicDecomposition.

BookProof.ChapterAtomicDecomposition.atomicPart_eq_sum_dirac (mu : Measure X) [SFinite mu] : atomicPart mu = Measure.sum (fun x : atoms mu => mu {(x : X)} • Measure.dirac (x : X))

Formalization note: Lean 4 identifier BookProof.ChapterAtomicDecomposition.atomicPart_eq_sum_dirac.

Preamble

-- Generated from ChapterAtomicDecomposition.lean — theorem BookProof.ChapterAtomicDecomposition.atomicPart_eq_sum_dirac
import Mathlib
import Definitions.Def_ChapterAtomicDecomposition
open BookProof.ChapterAtomicDecomposition


open MeasureTheory


variable {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]

Formal statement

theorem BookProof.ChapterAtomicDecomposition.atomicPart_eq_sum_dirac (mu : Measure X) [SFinite mu] :
    atomicPart mu = Measure.sum (fun x : atoms mu => mu {(x : X)} • Measure.dirac (x : X)) := by sorry

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