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Theorems · Theorem · probability

MeasureTheory.hasPDF_iff_of_aemeasurable

∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {x : MeasurableSpace Ω} {X : Ω → E}
  {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E},
  AEMeasurable X ℙ →
    (MeasureTheory.HasPDF X ℙ μ ↔
      (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧
        (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous μ)
Defined in
Mathlib.Probability.Density
Cited by
2 results in Mathlib
Foundations
Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
MeasurableSpace
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

ProbabilityTheory.IndepFun.add_hasPDF' · cited by 1IndepFun.add_hasPDF'ProbabilityTheory.IndepFun.mul_hasPDF' · cited by 1IndepFun.mul_hasPDF'MeasurableSpace · cited by 13106MeasurableSpaceMeasureTheory.Measure · cited by 10939MeasureTheory.MeasureMeasureTheory.Measure.map · cited by 858Measure.mapAEMeasurable · cited by 840AEMeasurableMeasureTheory.Measure.AbsolutelyContinuous · cited by 325Measure.AbsolutelyContinu…MeasureTheory.Measure.HaveLebesgueDecomposition · cited by 92Measure.HaveLebesgueDecom…MeasureTheory.HasPDF · cited by 37MeasureTheory.HasPDFMeasureTheory.hasPDF_iff · cited by 3MeasureTheory.hasPDF_iffMeasureTheory.hasPDF_iff_of_a…CITED BYCITES

Cites8

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Cited by2

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