Theorem: ProbabilityTheory.kernel.withDensity_rnDerivAux
File: Mathlib/Probability/Kernel/RadonNikodym.lean

tactic: ext a s hs
state_before: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
⊢ (withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) = κ
state_after: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) a) s = (κ a) s

tactic: rw [kernel.withDensity_apply']
state_before: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ((withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) a) s = (κ a) s
state_after: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (b : γ) in s, ↑(rnDerivAux κ (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s

case h.h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ Measurable (Function.uncurry fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal)

tactic: swap
state_before: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (b : γ) in s, ↑(rnDerivAux κ (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s

case h.h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ Measurable (Function.uncurry fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal)
state_after: case h.h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ Measurable (Function.uncurry fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal)

case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (b : γ) in s, ↑(rnDerivAux κ (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s

tactic: have : ∀ b, (Real.toNNReal b : ℝ≥0∞) = ENNReal.ofReal b := fun _ ↦ rfl
state_before: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (b : γ) in s, ↑(rnDerivAux κ (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s
state_after: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ∫⁻ (b : γ) in s, ↑(rnDerivAux κ (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s

tactic: simp_rw [this]
state_before: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ∫⁻ (b : γ) in s, ↑(rnDerivAux κ (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s
state_after: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ∫⁻ (b : γ) in s, ENNReal.ofReal (rnDerivAux κ (κ + η) a b) ∂(κ + η) a = (κ a) s

tactic: exact setLIntegral_rnDerivAux κ η a hs
state_before: case h.h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ∫⁻ (b : γ) in s, ENNReal.ofReal (rnDerivAux κ (κ + η) a b) ∂(κ + η) a = (κ a) s
state_after: no goals

tactic: exact (measurable_rnDerivAux _ _).ennreal_ofReal
state_before: case h.h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ Measurable (Function.uncurry fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal)
state_after: no goals