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`BookProof.ChapterAtomicDecomposition.not_continuousPart_zero_and_atoms_empty` (mu : Measure X) [IsProbabilityMeasure mu] : ¬ (continuousPart mu = 0 ∧ atoms mu = ∅)

Proved
BookProof.ChapterAtomicDecomposition.not_continuousPart_zero_and_atoms_empty

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAtomicDecomposition.

BookProof.ChapterAtomicDecomposition.not_continuousPart_zero_and_atoms_empty (mu : Measure X) [IsProbabilityMeasure mu] : ¬ (continuousPart mu = 0 ∧ atoms mu = ∅)

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

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


open MeasureTheory


variable {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]
Formal statement
theorem BookProof.ChapterAtomicDecomposition.not_continuousPart_zero_and_atoms_empty (mu : Measure X) [IsProbabilityMeasure mu] :
    ¬ (continuousPart mu = 0 ∧ atoms mu = ∅) := by sorry

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