Prove2Me

`BookProof.ChapterMixedPrior.exists_continuous_prior_beyond_atomic` (mu : Measure X) [IsProbabilityMeasure mu] (h : ¬ IsPurelyAtomic mu) : ∃ nu : Measure X, IsProbabilityMeasure nu

Proved
BookProof.ChapterMixedPrior.exists_continuous_prior_beyond_atomic

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMixedPrior.

BookProof.ChapterMixedPrior.exists_continuous_prior_beyond_atomic (mu : Measure X) [IsProbabilityMeasure mu] (h : ¬ IsPurelyAtomic mu) : ∃ nu : Measure X, IsProbabilityMeasure nu ∧ NullSingletonClass nu ∧ ¬ IsPurelyAtomic nu

Formalization note: Lean 4 identifier BookProof.ChapterMixedPrior.exists_continuous_prior_beyond_atomic.

Preamble
-- Generated from ChapterMixedPrior.lean — theorem BookProof.ChapterMixedPrior.exists_continuous_prior_beyond_atomic
import Definitions.Def_ChapterAtomicDecomposition
import Mathlib
import Definitions.Def_ChapterMixedPrior
open BookProof.ChapterMixedPrior


open MeasureTheory ProbabilityTheory


open BookProof.ChapterAtomicDecomposition

variable {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]
Formal statement
theorem BookProof.ChapterMixedPrior.exists_continuous_prior_beyond_atomic (mu : Measure X) [IsProbabilityMeasure mu]
    (h : ¬ IsPurelyAtomic mu) :
    ∃ nu : Measure X, IsProbabilityMeasure nu ∧ NullSingletonClass nu ∧ ¬ IsPurelyAtomic nu := by sorry

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