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

tactic: refine Submartingale.tendsto_snorm_one_limitProcess ?_ ?_
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ Tendsto
    (fun n =>
      snorm
        ((fun x => densityProcess κ ν n a x s) -
          Filtration.limitProcess (fun n x => densityProcess κ ν n a x s) (countableFiltration γ) (ν a))
        1 (ν a))
    atTop (𝓝 0)
state_after: case refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ Submartingale (fun n x => densityProcess κ ν n a x s) (countableFiltration γ) (ν a)

case refine_2
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ UniformIntegrable (fun n x => densityProcess κ ν n a x s) 1 (ν a)

tactic: exact (martingale_densityProcess hκν a hs).submartingale
state_before: case refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ Submartingale (fun n x => densityProcess κ ν n a x s) (countableFiltration γ) (ν a)
state_after: no goals

tactic: refine uniformIntegrable_of le_rfl ENNReal.one_ne_top ?_ ?_
state_before: case refine_2
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ UniformIntegrable (fun n x => densityProcess κ ν n a x s) 1 (ν a)
state_after: case refine_2.refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∀ (i : ℕ), AEStronglyMeasurable (fun x => densityProcess κ ν i a x s) (ν a)

case refine_2.refine_2
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∀ (ε : ℝ),
    0 < ε →
      ∃ C,
        ∀ (i : ℕ),
          snorm ({x | C ≤ ‖densityProcess κ ν i a x s‖₊}.indicator fun x => densityProcess κ ν i a x s) 1 (ν a) ≤
            ENNReal.ofReal ε

tactic: exact fun n ↦ (measurable_densityProcess_right κ ν n a hs).aestronglyMeasurable
state_before: case refine_2.refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∀ (i : ℕ), AEStronglyMeasurable (fun x => densityProcess κ ν i a x s) (ν a)
state_after: no goals

tactic: refine fun ε _ ↦ ⟨2, fun n ↦ le_of_eq_of_le ?_ (?_ : 0 ≤ ENNReal.ofReal ε)⟩
state_before: case refine_2.refine_2
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∀ (ε : ℝ),
    0 < ε →
      ∃ C,
        ∀ (i : ℕ),
          snorm ({x | C ≤ ‖densityProcess κ ν i a x s‖₊}.indicator fun x => densityProcess κ ν i a x s) 1 (ν a) ≤
            ENNReal.ofReal ε
state_after: case refine_2.refine_2.refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
⊢ snorm ({x | 2 ≤ ‖densityProcess κ ν n a x s‖₊}.indicator fun x => densityProcess κ ν n a x s) 1 (ν a) = 0

case refine_2.refine_2.refine_2
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
⊢ 0 ≤ ENNReal.ofReal ε

tactic: suffices {x | 2 ≤ ‖densityProcess κ ν n a x s‖₊} = ∅ by simp [this]
state_before: case refine_2.refine_2.refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
⊢ snorm ({x | 2 ≤ ‖densityProcess κ ν n a x s‖₊}.indicator fun x => densityProcess κ ν n a x s) 1 (ν a) = 0
state_after: case refine_2.refine_2.refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
⊢ {x | 2 ≤ ‖densityProcess κ ν n a x s‖₊} = ∅

tactic: ext x
state_before: case refine_2.refine_2.refine_1
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
⊢ {x | 2 ≤ ‖densityProcess κ ν n a x s‖₊} = ∅
state_after: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ x ∈ {x | 2 ≤ ‖densityProcess κ ν n a x s‖₊} ↔ x ∈ ∅

tactic: simp only [mem_setOf_eq, mem_empty_iff_false, iff_false, not_le]
state_before: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ x ∈ {x | 2 ≤ ‖densityProcess κ ν n a x s‖₊} ↔ x ∈ ∅
state_after: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ‖densityProcess κ ν n a x s‖₊ < 2

tactic: refine (?_ : _ ≤ (1 : ℝ≥0)).trans_lt one_lt_two
state_before: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ‖densityProcess κ ν n a x s‖₊ < 2
state_after: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ‖densityProcess κ ν n a x s‖₊ ≤ 1

tactic: rw [Real.nnnorm_of_nonneg (densityProcess_nonneg _ _ _ _ _ _)]
state_before: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ‖densityProcess κ ν n a x s‖₊ ≤ 1
state_after: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ⟨densityProcess κ ν n a x s, ⋯⟩ ≤ 1

tactic: exact mod_cast (densityProcess_le_one hκν _ _ _ _)
state_before: case refine_2.refine_2.refine_1.h
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ⟨densityProcess κ ν n a x s, ⋯⟩ ≤ 1
state_after: no goals

tactic: simp [this]
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
this : {x | 2 ≤ ‖densityProcess κ ν n a x s‖₊} = ∅
⊢ snorm ({x | 2 ≤ ‖densityProcess κ ν n a x s‖₊}.indicator fun x => densityProcess κ ν n a x s) 1 (ν a) = 0
state_after: no goals

tactic: simp
state_before: case refine_2.refine_2.refine_2
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
⊢ 0 ≤ ENNReal.ofReal ε
state_after: no goals