Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ
this :
  η.withDensity (κ.rnDeriv η) =
    ((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal).withDensity (κ.rnDeriv η)
⊢ (((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal * κ.rnDeriv η a x) a) s = (κ a) s

pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ
this :
  η.withDensity (κ.rnDeriv η) =
    ((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal).withDensity (κ.rnDeriv η)
⊢ Measurable (Function.uncurry fun a x ↦ (1 - κ.rnDerivAux (κ + η) a x).toNNReal)

pp: case hg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ
this :
  η.withDensity (κ.rnDeriv η) =
    ((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal).withDensity (κ.rnDeriv η)
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

Tactic:
Kernel.withDensity_apply'

Goal state after:
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ
this :
  η.withDensity (κ.rnDeriv η) =
    ((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal).withDensity (κ.rnDeriv η)
⊢ ∫⁻ (b : γ) in s, ↑(1 - κ.rnDerivAux (κ + η) a b).toNNReal * κ.rnDeriv η a b ∂(κ + η) a = (κ a) s

case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ
this :
  η.withDensity (κ.rnDeriv η) =
    ((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal).withDensity (κ.rnDeriv η)
⊢ Measurable (Function.uncurry fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal * κ.rnDeriv η a x)

case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ
this :
  η.withDensity (κ.rnDeriv η) =
    ((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal).withDensity (κ.rnDeriv η)
⊢ Measurable (Function.uncurry fun a x ↦ (1 - κ.rnDerivAux (κ + η) a x).toNNReal)

case hg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ
this :
  η.withDensity (κ.rnDeriv η) =
    ((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal).withDensity (κ.rnDeriv η)
⊢ Measurable (Function.uncurry (κ.rnDeriv η))