Theorems · Theorem · probability

ProbabilityTheory.Kernel.integral_integral_add_comp

∀ {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}
  {η : ProbabilityTheory.Kernel β γ} [inst_1 : NormedSpace ℝ E] ⦃f g : γ → E⦄,
  MeasureTheory.Integrable f ((η.comp κ) a) →
    MeasureTheory.Integrable g ((η.comp κ) a) →
      ∫ (x : β), ∫ (y : γ), f y + g y ∂η x ∂κ a =
        ∫ (x : β), ∫ (y : γ), f y ∂η x ∂κ a + ∫ (x : β), ∫ (y : γ), g y ∂η x ∂κ a

Defined in

Mathlib.Probability.Kernel.Composition.IntegralCompProd

Cited by

1 results in Mathlib

Foundations

Depth 260 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace