Theorems · Theorem · probability

MeasureTheory.pdf.eq_of_map_eq_withDensity

∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω}
  {μ : MeasureTheory.Measure E} [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ]
  (f : E → ENNReal),
  AEMeasurable f μ → (MeasureTheory.Measure.map X ℙ = μ.withDensity f ↔ MeasureTheory.pdf X ℙ μ =ᵐ[μ] f)

Defined in: Mathlib.Probability.Density
Cited by: 2 results in Mathlib
Foundations: Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasurableSpace MeasureTheory.IsFiniteMeasure MeasureTheory.HasPDF

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Cites19

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement and proof · cited by 9,879
Top.topproof · cited by 9,680
Set.univproof · cited by 3,945
MeasureTheory.aestatement and proof · cited by 2,352
Filter.EventuallyEqstatement and proof · cited by 1,912
MeasureTheory.IsFiniteMeasurestatement and proof · cited by 1,078
MeasureTheory.Measure.mapstatement · cited by 858
AEMeasurablestatement and proof · cited by 840
Measurable.aemeasurableproof · cited by 304
MeasureTheory.Measure.withDensitystatement and proof · cited by 265

Cited by2

Results whose statement or proof uses this declaration.

MeasureTheory.pdf.IsUniform.pdf_eqproof · cited by 3
MeasureTheory.pdf.indepFun_iff_pdf_prod_eq_pdf_mul_pdfproof · cited by 0