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

tactic: refine tendsto_of_integral_tendsto_of_monotone ?_ (integrable_const _) ?_ ?_ ?_
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ∀ᵐ (x : γ) ∂(fst κ) a, Tendsto (fun m => density κ (fst κ) a x (seq m)) atTop (𝓝 1)
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ∀ (n : ℕ), Integrable (fun x => density κ (fst κ) a x (seq n)) ((fst κ) 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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun i => ∫ (a_1 : γ), density κ (fst κ) a a_1 (seq i) ∂(fst κ) a) atTop (𝓝 (∫ (a : γ), 1 ∂(fst κ) a))

case refine_3
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ∀ᵐ (a_1 : γ) ∂(fst κ) a, Monotone fun i => density κ (fst κ) a a_1 (seq i)

case refine_4
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ∀ᵐ (a_1 : γ) ∂(fst κ) a, ∀ (i : ℕ), density κ (fst κ) a a_1 (seq i) ≤ 1

tactic: exact fun m ↦ integrable_density le_rfl _ (hseq_meas m)
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ∀ (n : ℕ), Integrable (fun x => density κ (fst κ) a x (seq n)) ((fst κ) a)
state_after: no goals

tactic: rw [MeasureTheory.integral_const, smul_eq_mul, mul_one]
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun i => ∫ (a_1 : γ), density κ (fst κ) a a_1 (seq i) ∂(fst κ) a) atTop (𝓝 (∫ (a : γ), 1 ∂(fst κ) a))
state_after: 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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun i => ∫ (a_1 : γ), density κ (fst κ) a a_1 (seq i) ∂(fst κ) a) atTop (𝓝 (((fst κ) a) univ).toReal)

tactic: convert tendsto_integral_density_of_monotone (κ := κ) le_rfl a seq hseq hseq_iUnion hseq_meas
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun i => ∫ (a_1 : γ), density κ (fst κ) a a_1 (seq i) ∂(fst κ) a) atTop (𝓝 (((fst κ) a) univ).toReal)
state_after: case h.e'_5.h.e'_3.h.e'_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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ((fst κ) a) univ = (κ a) univ

tactic: rw [fst_apply' _ _ MeasurableSet.univ]
state_before: case h.e'_5.h.e'_3.h.e'_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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ((fst κ) a) univ = (κ a) univ
state_after: case h.e'_5.h.e'_3.h.e'_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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ (κ a) {p | p.1 ∈ univ} = (κ a) univ

tactic: simp only [mem_univ, setOf_true]
state_before: case h.e'_5.h.e'_3.h.e'_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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ (κ a) {p | p.1 ∈ univ} = (κ a) univ
state_after: no goals

tactic: exact ae_of_all _ (fun c n m hnm ↦ density_mono_set le_rfl a c (hseq hnm))
state_before: case refine_3
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ∀ᵐ (a_1 : γ) ∂(fst κ) a, Monotone fun i => density κ (fst κ) a a_1 (seq i)
state_after: no goals

tactic: exact ae_of_all _ (fun x m ↦ density_le_one le_rfl a x (seq m))
state_before: case refine_4
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)
inst✝ : IsFiniteKernel κ
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ ∀ᵐ (a_1 : γ) ∂(fst κ) a, ∀ (i : ℕ), density κ (fst κ) a a_1 (seq i) ≤ 1
state_after: no goals