Theorems · Theorem · probability

ProbabilityTheory.Kernel.indepFun_iff_compProd_map_prod_eq_compProd_prod_map_map

∀ {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {γ : Type u_6} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω}
  {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
  [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] {f : Ω → β} {g : Ω → γ},
  Measurable f →
    Measurable g →
      (ProbabilityTheory.Kernel.IndepFun f g κ μ ↔
        μ.compProd (κ.map fun ω => (f ω, g ω)) = μ.compProd ((κ.map f).prod (κ.map g)))

Two random variables f, g are independent given a kernel κ and a measure μ iff μ ⊗ₘ κ.map (fun ω ↦ (f ω, g ω)) = μ ⊗ₘ (κ.map f ×ₖ κ.map g).

Defined in

Mathlib.Probability.Independence.Kernel.IndepFun

Cited by

1 results in Mathlib

Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasureProbabilityTheory.IsFiniteKernel