Prove2Me

`BookProof.ChapterSelectingEvents.exists_continuous_atomic_decomposition` [MeasurableSingletonClass X] : ∃ cont atom : Measure X, μ = cont + atom ∧ NullSingletonClass cont ∧ ∃ A :

Proved
BookProof.ChapterSelectingEvents.exists_continuous_atomic_decomposition

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSelectingEvents.

BookProof.ChapterSelectingEvents.exists_continuous_atomic_decomposition [MeasurableSingletonClass X] : ∃ cont atom : Measure X, μ = cont + atom ∧ NullSingletonClass cont ∧ ∃ A : Set X, A.Countable ∧ atom Aᶜ = 0

Formalization note: Lean 4 identifier BookProof.ChapterSelectingEvents.exists_continuous_atomic_decomposition.

Preamble
-- Generated from ChapterSelectingEvents.lean — theorem BookProof.ChapterSelectingEvents.exists_continuous_atomic_decomposition
import Mathlib
import Definitions.Def_ChapterSelectingEvents
open BookProof.ChapterSelectingEvents


open scoped BigOperators
open MeasureTheory ProbabilityTheory


variable (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

variable {Ω Data : Type*} [MeasurableSpace Ω]
variable {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ]
Formal statement
theorem BookProof.ChapterSelectingEvents.exists_continuous_atomic_decomposition [MeasurableSingletonClass X] :
    ∃ cont atom : Measure X, μ = cont + atom ∧ NullSingletonClass cont ∧
      ∃ A : Set X, A.Countable ∧ atom Aᶜ = 0 := by sorry

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