Theorems · Theorem · probability

MeasureTheory.Measure.setIntegral_compProd

∀ {α : Type u_5} {β : Type u_6} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β}
  {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {E : Type u_8}
  [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {s : Set α},
  MeasurableSet s →
    ∀ {t : Set β},
      MeasurableSet t →
        ∀ {f : α × β → E},
          MeasureTheory.IntegrableOn f (s ×ˢ t) (μ.compProd κ) →
            ∫ (x : α × β) in s ×ˢ t, f x ∂μ.compProd κ = ∫ (a : α) in s, ∫ (b : β) in t, f (a, b) ∂κ a ∂μ

Defined in

Mathlib.Probability.Kernel.Composition.IntegralCompProd

Cited by

1 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SFiniteProbabilityTheory.IsSFiniteKernelNormedAddCommGroupNormedSpace