LeanMachineLearning exposition

ProbabilityTheory.Kernel.sub_of_isSFiniteKernel🔗

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Minimal Lean file

sub_of_isSFiniteKernel🔗

LemmaProbabilityTheory.Kernel.sub_of_isSFiniteKernel
Details

No docstring.

🔗theorem
ProbabilityTheory.Kernel.sub_of_isSFiniteKernel.{u_1, u_2}
  {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated α β] {κ η : Kernel α β}
  [IsSFiniteKernel κ] [IsSFiniteKernel η]
  [(η : Kernel α β) → Decidable (IsSFiniteKernel η)] :
  κ - η = (withDensity η fun a => rnDeriv κ η a - 1) + singularPart κ η
ProbabilityTheory.Kernel.sub_of_isSFiniteKernel.{u_1,
    u_2}
  {α : Type u_1} {β : Type u_2}
  {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated
      α β]
  {κ η : Kernel α β} [IsSFiniteKernel κ]
  [IsSFiniteKernel η]
  [(η : Kernel α β) →
      Decidable (IsSFiniteKernel η)] :
  κ - η =
    (withDensity η fun a =>
        rnDeriv κ η a - 1) +
      singularPart κ η

Code

lemma sub_of_isSFiniteKernel [IsSFiniteKernel κ] [IsSFiniteKernel η]
    [∀ η : Kernel α β, Decidable (IsSFiniteKernel η)] :
    κ - η = η.withDensity (fun a ↦ κ.rnDeriv η a - 1) + κ.singularPart η
Type uses (1)
Body uses (1)
Used by (1)

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Proof
by
  rw [sub_def, dif_pos]
  exact ⟨inferInstance, inferInstance⟩

Dependency graph

Type dependencies (1)

instSubOfDecidableIsSFiniteKernel_leanMachineLearning🔗

InstanceProbabilityTheory.Kernel.instSubOfDecidableIsSFiniteKernel_leanMachineLearning
Details

No docstring.

🔗def
ProbabilityTheory.Kernel.instSubOfDecidableIsSFiniteKernel_leanMachineLearning.{u_1,
    u_2}
  {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated α β]
  [(η : Kernel α β) → Decidable (IsSFiniteKernel η)] : Sub (Kernel α β)
ProbabilityTheory.Kernel.instSubOfDecidableIsSFiniteKernel_leanMachineLearning.{u_1,
    u_2}
  {α : Type u_1} {β : Type u_2}
  {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated
      α β]
  [(η : Kernel α β) →
      Decidable (IsSFiniteKernel η)] :
  Sub (Kernel α β)

Code

noncomputable
instance [∀ η : Kernel α β, Decidable (IsSFiniteKernel η)] :
    Sub (Kernel α β) where
  sub κ η
Used by (11)

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Proof
if h : IsSFiniteKernel κ ∧ IsSFiniteKernel η
    then
      have := h.1
      have := h.2
      η.withDensity (fun a ↦ κ.rnDeriv η a - 1) + κ.singularPart η
    else 0