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Theorems · Theorem · probability

Real.hasPDF_iff

∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → ℝ} [MeasureTheory.SFinite ℙ],
  MeasureTheory.HasPDF X ℙ MeasureTheory.volume ↔
    AEMeasurable X ℙ ∧ (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous MeasureTheory.volume
Defined in
Mathlib.Probability.Density
Cited by
1 results in Mathlib
Foundations
Depth 243 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
MeasureTheory.SFinite
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Real.hasPDF_iff_of_aemeasurable · cited by 0Real.hasPDF_iff_of_aemeas…Real · cited by 25697RealMeasurableSpace · cited by 13106MeasurableSpaceMeasureTheory.Measure · cited by 10939MeasureTheory.MeasureMeasureTheory.MeasureSpace.volume · cited by 1323MeasureSpace.volumeMeasureTheory.Measure.map · cited by 858Measure.mapAEMeasurable · cited by 840AEMeasurableMeasureTheory.SFinite · cited by 449MeasureTheory.SFiniteMeasureTheory.Measure.AbsolutelyContinuous · cited by 325Measure.AbsolutelyContinu…MeasureTheory.HasPDF · cited by 37MeasureTheory.HasPDFMeasureTheory.hasPDF_iff · cited by 3MeasureTheory.hasPDF_iffReal.hasPDF_iffCITED BYCITES

Cites10

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Cited by1

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