Theorem: ProbabilityTheory.Kernel.tendsto_densityProcess_atTop_empty_of_antitone
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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun m => κ.densityProcess ν n a x (seq m)) atTop (𝓝 (κ.densityProcess ν n a x ∅))
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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun m => ((κ a) (countablePartitionSet n x ×ˢ seq m) / (ν a) (countablePartitionSet n x)).toReal) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x)).toReal)
tactic: simp_rw [densityProcess]
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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun m => ((κ a) (countablePartitionSet n x ×ˢ seq m) / (ν a) (countablePartitionSet n x)).toReal) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x)).toReal)
state_after: case pos
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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : (ν a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => ((κ a) (countablePartitionSet n x ×ˢ seq m) / (ν a) (countablePartitionSet n x)).toReal) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x)).toReal)

case neg
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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => ((κ a) (countablePartitionSet n x ×ˢ seq m) / (ν a) (countablePartitionSet n x)).toReal) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x)).toReal)
tactic: by_cases h0 : ν a (countablePartitionSet n x) = 0
state_before: case neg
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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => ((κ a) (countablePartitionSet n x ×ˢ seq m) / (ν a) (countablePartitionSet n x)).toReal) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x)).toReal)
state_after: case neg.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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ (κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x) ≠ ⊤

case neg.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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / (ν a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x)))
tactic: refine (ENNReal.tendsto_toReal ?_).comp ?_
state_before: case neg.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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / (ν a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅) / (ν a) (countablePartitionSet n x)))
state_after: case neg.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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅)))

case neg.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
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ (κ a) (countablePartitionSet n x ×ˢ ∅) ≠ 0 ∨ (ν a) (countablePartitionSet n x) ≠ 0
tactic: refine ENNReal.Tendsto.div_const ?_ ?_