Theorems · Theorem · probability

MeasureTheory.pdf.lintegral_eq_measure_univ

∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω}
  {μ : MeasureTheory.Measure E} {X : Ω → E} [MeasureTheory.HasPDF X ℙ μ],
  ∫⁻ (x : E), MeasureTheory.pdf X ℙ μ x ∂μ = ℙ Set.univ

Defined in: Mathlib.Probability.Density
Cited by: 1 results in Mathlib
Foundations: Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasurableSpace MeasureTheory.HasPDF

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Cites15

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

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement and proof · cited by 9,879
Set.univstatement and proof · cited by 3,945
MeasureTheory.lintegralstatement · cited by 1,152
MeasurableSet.univproof · cited by 178
Set.preimage_univproof · cited by 46
MeasureTheory.HasPDFstatement and proof · cited by 37
MeasureTheory.Measure.map_apply_of_aemeasurableproof · cited by 36
MeasureTheory.pdfstatement and proof · cited by 32

Cited by1

Results whose statement or proof uses this declaration.

MeasureTheory.pdf.eq_of_map_eq_withDensityproof · cited by 2