Theorems · Theorem · probability

ProbabilityTheory.setIntegral_compProd

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {E : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} [inst : NormedAddCommGroup E] {a : α} {κ : ProbabilityTheory.Kernel α β}
  [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η]
  [inst_3 : NormedSpace ℝ E] {f : β × γ → E} {s : Set β} {t : Set γ},
  MeasurableSet s →
    MeasurableSet t →
      MeasureTheory.IntegrableOn f (s ×ˢ t) ((κ.compProd η) a) →
        ∫ (z : β × γ) in s ×ˢ t, f z ∂(κ.compProd η) a = ∫ (x : β) in s, ∫ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a

Defined in

Mathlib.Probability.Kernel.Composition.IntegralCompProd

Cited by

4 results in Mathlib

Foundations

Depth 263 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupProbabilityTheory.IsSFiniteKernelProbabilityTheory.IsSFiniteKernelNormedSpace