-- Mathlib/Probability/Kernel/Disintegration/Density.lean
lemma setLIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A) :
    ∫⁻ x in A, ENNReal.ofReal (density κ ν a x s) ∂(ν a) = κ a (A ×ˢ s)

Proof state:
case hfi
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ Integrable (fun x => κ.density ν a x s) ((ν a).restrict A)

Tactic:
exact (integrable_density hκν a hs).restrict

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
⊢ ∫⁻ (x : γ) in A, ENNReal.ofReal (κ.density ν a x s) ∂ν a = (κ a) (A ×ˢ s)

Tactic:
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)

Result:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ∫⁻ (x : γ) in A, ENNReal.ofReal (κ.density ν a x s) ∂ν a = (κ a) (A ×ˢ s)

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ∫⁻ (x : γ) in A, ENNReal.ofReal (κ.density ν a x s) ∂ν a = (κ a) (A ×ˢ s)

Tactic:
rw [← ofReal_integral_eq_lintegral_ofReal]

Result:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ENNReal.ofReal (∫ (x : γ) in A, κ.density ν a x s ∂ν a) = (κ a) (A ×ˢ s)

case hfi
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ Integrable (fun x => κ.density ν a x s) ((ν a).restrict A)

case f_nn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ 0 ≤ᶠ[ae ((ν a).restrict A)] fun x => κ.density ν a x s

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ENNReal.ofReal (∫ (x : γ) in A, κ.density ν a x s ∂ν a) = (κ a) (A ×ˢ s)

Tactic:
rw [setIntegral_density hκν a hs hA,
  ENNReal.ofReal_toReal (measure_ne_top _ _)]

Result:
no goals

Proof state:
case f_nn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ 0 ≤ᶠ[ae ((ν a).restrict A)] fun x => κ.density ν a x s

Tactic:
exact ae_of_all _ (fun _ ↦ density_nonneg hκν _ _ _)

Result:
no goals