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

tactic: have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun m => ∫ (x : γ), density κ ν a x (seq m) ∂ν a) atTop (𝓝 ((κ a) univ).toReal)
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ Tendsto (fun m => ∫ (x : γ), density κ ν a x (seq m) ∂ν a) atTop (𝓝 ((κ a) univ).toReal)

tactic: simp_rw [integral_density hκν a (hseq_meas _)]
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ Tendsto (fun m => ∫ (x : γ), density κ ν a x (seq m) ∂ν a) atTop (𝓝 ((κ a) univ).toReal)
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 : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ Tendsto (fun m => ((κ a) (univ ×ˢ seq m)).toReal) atTop (𝓝 ((κ a) univ).toReal)

tactic: have h_cont := ENNReal.continuousOn_toReal.continuousAt (x := κ a univ) ?_
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ Tendsto (fun m => ((κ a) (univ ×ˢ seq m)).toReal) atTop (𝓝 ((κ a) univ).toReal)
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)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Tendsto (fun m => ((κ a) (univ ×ˢ seq m)).toReal) atTop (𝓝 ((κ a) univ).toReal)

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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ {a | a ≠ ⊤} ∈ 𝓝 ((κ a) univ)

tactic: swap
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Tendsto (fun m => ((κ a) (univ ×ˢ seq m)).toReal) atTop (𝓝 ((κ a) univ).toReal)

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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ {a | a ≠ ⊤} ∈ 𝓝 ((κ a) univ)
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ {a | a ≠ ⊤} ∈ 𝓝 ((κ a) univ)

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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Tendsto (fun m => ((κ a) (univ ×ˢ seq m)).toReal) atTop (𝓝 ((κ a) univ).toReal)

tactic: refine h_cont.tendsto.comp ?_
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Tendsto (fun m => ((κ a) (univ ×ˢ seq m)).toReal) atTop (𝓝 ((κ a) univ).toReal)
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)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Tendsto (fun m => (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) univ))

tactic: have h := tendsto_measure_iUnion (s := fun m ↦ univ ×ˢ seq m) (μ := κ a) ?_
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Tendsto (fun m => (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) univ))
state_after: 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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ Tendsto (fun m => (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) univ))

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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Monotone fun m => univ ×ˢ seq m

tactic: swap
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ Tendsto (fun m => (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) univ))

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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Monotone fun m => univ ×ˢ seq m
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Monotone fun m => univ ×ˢ seq m

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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ Tendsto (fun m => (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) univ))

tactic: convert h
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ Tendsto (fun m => (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) univ))
state_after: case h.e'_5.h.e'_3.h.e'_6
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ univ = ⋃ n, univ ×ˢ seq n

tactic: rw [← prod_iUnion, hseq_iUnion]
state_before: case h.e'_5.h.e'_3.h.e'_6
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ univ = ⋃ n, univ ×ˢ seq n
state_after: case h.e'_5.h.e'_3.h.e'_6
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ univ = univ ×ˢ univ

tactic: simp only [univ_prod_univ, measure_univ]
state_before: case h.e'_5.h.e'_3.h.e'_6
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))
⊢ univ = univ ×ˢ univ
state_after: no goals

tactic: rw [mem_nhds_iff]
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ {a | a ≠ ⊤} ∈ 𝓝 ((κ a) univ)
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ ∃ t ⊆ {a | a ≠ ⊤}, IsOpen t ∧ (κ a) univ ∈ t

tactic: refine ⟨Iio (κ a univ + 1), fun x hx ↦ ne_top_of_lt (?_ : x < κ a univ + 1), isOpen_Iio, ?_⟩
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ ∃ t ⊆ {a | a ≠ ⊤}, IsOpen t ∧ (κ a) univ ∈ t
state_after: case refine_1.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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
x : ℝ≥0∞
hx : x ∈ Iio ((κ a) univ + 1)
⊢ x < (κ a) univ + 1

case refine_1.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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ (κ a) univ ∈ Iio ((κ a) univ + 1)

tactic: simpa using hx
state_before: case refine_1.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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
x : ℝ≥0∞
hx : x ∈ Iio ((κ a) univ + 1)
⊢ x < (κ a) univ + 1
state_after: no goals

tactic: simp only [mem_Iio]
state_before: case refine_1.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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ (κ a) univ ∈ Iio ((κ a) univ + 1)
state_after: case refine_1.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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ (κ a) univ < (κ a) univ + 1

tactic: exact ENNReal.lt_add_right (measure_ne_top _ _) one_ne_zero
state_before: case refine_1.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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ (κ a) univ < (κ a) univ + 1
state_after: no goals

tactic: intro n m hnm x
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Monotone fun m => univ ×ˢ seq m
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
n m : ℕ
hnm : n ≤ m
x : γ × β
⊢ x ∈ (fun m => univ ×ˢ seq m) n → x ∈ (fun m => univ ×ˢ seq m) m

tactic: simp only [mem_prod, mem_univ, true_and]
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
n m : ℕ
hnm : n ≤ m
x : γ × β
⊢ x ∈ (fun m => univ ×ˢ seq m) n → x ∈ (fun m => univ ×ˢ seq m) m
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
n m : ℕ
hnm : n ≤ m
x : γ × β
⊢ x.2 ∈ seq n → x.2 ∈ seq m

tactic: exact fun h ↦ hseq hnm h
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
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
n m : ℕ
hnm : n ≤ m
x : γ × β
⊢ x.2 ∈ seq n → x.2 ∈ seq m
state_after: no goals