Theorems · Theorem · probability

MeasureTheory.Measure.compProd_withDensity

∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α}
  {κ : ProbabilityTheory.Kernel α β} {f : α → β → ENNReal} [MeasureTheory.SFinite μ]
  [inst : ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel (κ.withDensity f)],
  Measurable (Function.uncurry f) → μ.compProd (κ.withDensity f) = (μ.compProd κ).withDensity fun p => f p.1 p.2

A composition-product of a measure with a kernel defined with withDensity is equal to the withDensity of the composition-product.

Defined in: Mathlib.Probability.Kernel.CompProdEqIff
Cited by: 1 results in Mathlib
Foundations: Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasureTheory.SFinite ProbabilityTheory.IsSFiniteKernel ProbabilityTheory.IsSFiniteKernel

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

DFunLike.coeproof · cited by 62,936
Setproof · 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.preimageproof · cited by 4,946
MeasurableSetproof · cited by 3,075
Measurablestatement and proof · cited by 1,499
ProbabilityTheory.Kernelstatement and proof · cited by 1,281
MeasureTheory.lintegralproof · cited by 1,152
Set.indicatorproof · cited by 723
MeasureTheory.SFinitestatement and proof · cited by 449

Cited by1

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ProbabilityTheory.Kernel.ae_eq_of_compProd_eqproof · cited by 3