Theorem: ProbabilityTheory.Kernel.tendsto_setIntegral_densityProcess
File: Mathlib/Probability/Kernel/Disintegration/Density.lean

Proof steps:
state_before: 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
⊢ Tendsto (fun i => ∫ (x : γ) in A, κ.densityProcess ν i a x s ∂ν a) atTop (𝓝 (∫ (x : γ) in A, κ.density ν a x s ∂ν a))
state_after: 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
⊢ Tendsto (fun i => eLpNorm ((fun i x => κ.densityProcess ν i a x s) i - fun x => κ.density ν a x s) 1 (ν a)) atTop
    (𝓝 0)
tactic: refine tendsto_setIntegral_of_L1' (μ := ν a) (fun x ↦ density κ ν a x s)
    (integrable_density hκν a hs) (F := fun i x ↦ densityProcess κ ν i a x s) (l := atTop)
    (eventually_of_forall (fun n ↦ integrable_densityProcess hκν _ _ hs)) ?_ A
state_before: 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
⊢ Tendsto (fun i => eLpNorm ((fun i x => κ.densityProcess ν i a x s) i - fun x => κ.density ν a x s) 1 (ν a)) atTop
    (𝓝 0)
state_after: 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
n : ℕ
⊢ eLpNorm
      ((fun x => κ.densityProcess ν n a x s) -
        Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s) (countableFiltration γ) (ν a))
      1 (ν a) =
    eLpNorm ((fun i x => κ.densityProcess ν i a x s) n - fun x => κ.density ν a x s) 1 (ν a)
tactic: refine (tendsto_congr fun n ↦ ?_).mp (tendsto_eLpNorm_one_densityProcess_limitProcess hκν a hs)
state_before: 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
n : ℕ
⊢ eLpNorm
      ((fun x => κ.densityProcess ν n a x s) -
        Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s) (countableFiltration γ) (ν a))
      1 (ν a) =
    eLpNorm ((fun i x => κ.densityProcess ν i a x s) n - fun x => κ.density ν a x s) 1 (ν a)
state_after: 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
n : ℕ
⊢ (fun x => κ.densityProcess ν n a x s) -
      Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s) (countableFiltration γ) (ν a) =ᶠ[ae (ν a)]
    (fun i x => κ.densityProcess ν i a x s) n - fun x => κ.density ν a x s
tactic: refine eLpNorm_congr_ae ?_
state_before: 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
n : ℕ
⊢ (fun x => κ.densityProcess ν n a x s) -
      Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s) (countableFiltration γ) (ν a) =ᶠ[ae (ν a)]
    (fun i x => κ.densityProcess ν i a x s) n - fun x => κ.density ν a x s
state_after: no goals
tactic: exact EventuallyEq.rfl.sub (density_ae_eq_limitProcess hκν a hs).symm