l3lab/ntp-mathlib huggingface dataset:
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
p q : β
⊢ ∃ u v,
    MeasurableSet u ∧
      MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u ∩ v) = 0)

Next tactic:
by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · 

Proof state:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
p q : β
H : p ∈ s ∧ q ∈ s ∧ p < q
⊢ ∃ u v,
    MeasurableSet u ∧
      MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u ∩ v) = 0)

Next tactic:
rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      

Proof state:
case pos.intro.intro.intro.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
p q : β
H : p ∈ s ∧ q ∈ s ∧ p < q
u v : Set α
hu : MeasurableSet u
hv : MeasurableSet v
h'u : {x | f x < p} ⊆ u
h'v : {x | q < f x} ⊆ v
hμ : ↑↑μ (u ∩ v) = 0
⊢ ∃ u v,
    MeasurableSet u ∧
      MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u ∩ v) = 0)

Next tactic:
exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · 

Proof state:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
p q : β
H : ¬(p ∈ s ∧ q ∈ s ∧ p < q)
⊢ ∃ u v,
    MeasurableSet u ∧
      MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u ∩ v) = 0)

Next tactic:
refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
      

Proof state:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
p q : β
H : ¬(p ∈ s ∧ q ∈ s ∧ p < q)
ps : p ∈ s
qs : q ∈ s
pq : p < q
⊢ ↑↑μ (univ ∩ univ) = 0

Next tactic:
simp only [not_and] at H
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
      simp only [not_and] at H
      

Proof state:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
p q : β
ps : p ∈ s
qs : q ∈ s
pq : p < q
H : p ∈ s → q ∈ s → ¬p < q
⊢ ↑↑μ (univ ∩ univ) = 0

Next tactic:
exact (H ps qs pq).elim
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
      simp only [not_and] at H
      exact (H ps qs pq).elim
  

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
h' :
  ∀ (p q : β),
    ∃ u v,
      MeasurableSet u ∧
        MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u ∩ v) = 0)
⊢ AEMeasurable f

Next tactic:
choose! u v huv using h'
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
      simp only [not_and] at H
      exact (H ps qs pq).elim
  choose! u v huv using h'
  

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u p q ∩ v p q) = 0)
⊢ AEMeasurable f

Next tactic:
let u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
      simp only [not_and] at H
      exact (H ps qs pq).elim
  choose! u v huv using h'
  let u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
  

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
⊢ AEMeasurable f

Next tactic:
have u'_meas : ∀ i, MeasurableSet (u' i) := by
    intro i
    exact MeasurableSet.biInter (s_count.mono (inter_subset_left _ _)) fun b _ => (huv i b).1
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
      simp only [not_and] at H
      exact (H ps qs pq).elim
  choose! u v huv using h'
  let u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
  have u'_meas : ∀ i, MeasurableSet (u' i) := by
    

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
⊢ ∀ (i : β), MeasurableSet (u' i)

Next tactic:
intro i
/-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere
measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
    {m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
    [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
    [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
    (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
    AEMeasurable f μ := by
  haveI : Encodable s := s_count.toEncodable
  have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
      { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
    intro p q
    by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
    · rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
      exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
    · refine'
        ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
          fun ps qs pq => _⟩
      simp only [not_and] at H
      exact (H ps qs pq).elim
  choose! u v huv using h'
  let u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
  have u'_meas : ∀ i, MeasurableSet (u' i) := by
    intro i
    

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : Set.Countable s
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ ↑↑μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → ↑↑μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
i : β
⊢ MeasurableSet (u' i)

Next tactic:
exact MeasurableSet.biInter (s_count.mono (inter_subset_left _ _)) fun b _ => (huv i b).1