Theorem: MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets
File: Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean

tactic: haveI : Encodable s := s_count.toEncodable
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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

tactic: choose! u v huv using h'
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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

tactic: let u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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

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
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
⊢ AEMeasurable f

tactic: let f' : α → β := fun x => ⨅ i : s, piecewise (u' i) (fun _ => (i : β)) (fun _ => (⊤ : β)) x
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
⊢ AEMeasurable f

tactic: have f'_meas : Measurable f' := by
  apply measurable_iInf
  exact fun i => Measurable.piecewise (u'_meas i) measurable_const measurable_const
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
⊢ AEMeasurable f

tactic: let t := ⋃ (p : s) (q : ↥(s ∩ Ioi p)), u' p ∩ v p q
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
⊢ AEMeasurable f

tactic: have μt : μ t ≤ 0 :=
  calc
    μ t ≤ ∑' (p : s) (q : ↥(s ∩ Ioi p)), μ (u' p ∩ v p q) := by
      refine (measure_iUnion_le _).trans ?_
      refine ENNReal.tsum_le_tsum fun p => ?_
      refine @measure_iUnion_le _ _ _ _ ?_ _
      exact (s_count.mono (inter_subset_left _ _)).to_subtype
    _ ≤ ∑' (p : s) (q : ↥(s ∩ Ioi p)), μ (u p q ∩ v p q) := by
      refine ENNReal.tsum_le_tsum fun p => ?_
      refine ENNReal.tsum_le_tsum fun q => measure_mono ?_
      exact inter_subset_inter_left _ (biInter_subset_of_mem q.2)
    _ = ∑' (p : s) (_ : ↥(s ∩ Ioi p)), (0 : ℝ≥0∞) := by
      congr
      ext1 p
      congr
      ext1 q
      exact (huv p q).2.2.2.2 p.2 q.2.1 q.2.2
    _ = 0 := by simp only [tsum_zero]
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
⊢ AEMeasurable f

tactic: exact ⟨f', f'_meas, ff'⟩
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
ff' : ∀ᵐ (x : α) ∂μ, f x = f' x
⊢ AEMeasurable f

tactic: intro p q
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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)

tactic: by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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)

tactic: rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v, hμ⟩
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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)

tactic: exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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)

tactic: refine'
  ⟨univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _,
    fun ps qs pq => _⟩
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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)

tactic: simp only [not_and] at H
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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

tactic: exact (H ps qs pq).elim
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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

tactic: intro i
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
⊢ ∀ (i : β), MeasurableSet (u' i)

tactic: exact MeasurableSet.biInter (s_count.mono (inter_subset_left _ _)) fun b _ => (huv i b).1
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
i : β
⊢ MeasurableSet (u' i)

tactic: apply measurable_iInf
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
⊢ Measurable f'

tactic: exact fun i => Measurable.piecewise (u'_meas i) measurable_const measurable_const
state_before: case hf
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
⊢ ∀ (i : ↑s), Measurable fun b => piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) b

tactic: refine (measure_iUnion_le _).trans ?_
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
⊢ ↑↑μ t ≤ ∑' (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u' ↑p ∩ v ↑p ↑q)

tactic: refine ENNReal.tsum_le_tsum fun p => ?_
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
⊢ (∑' (i : ↑s), ↑↑μ (⋃ (q : ↑(s ∩ Ioi ↑i)), u' ↑i ∩ v ↑i ↑q)) ≤ ∑' (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u' ↑p ∩ v ↑p ↑q)

tactic: refine @measure_iUnion_le _ _ _ _ ?_ _
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
p : ↑s
⊢ ↑↑μ (⋃ (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q) ≤ ∑' (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u' ↑p ∩ v ↑p ↑q)

tactic: exact (s_count.mono (inter_subset_left _ _)).to_subtype
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
p : ↑s
⊢ Countable ↑(s ∩ Ioi ↑p)

tactic: refine ENNReal.tsum_le_tsum fun p => ?_
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
⊢ (∑' (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u' ↑p ∩ v ↑p ↑q)) ≤ ∑' (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u ↑p ↑q ∩ v ↑p ↑q)

tactic: refine ENNReal.tsum_le_tsum fun q => measure_mono ?_
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
p : ↑s
⊢ (∑' (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u' ↑p ∩ v ↑p ↑q)) ≤ ∑' (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u ↑p ↑q ∩ v ↑p ↑q)

tactic: exact inter_subset_inter_left _ (biInter_subset_of_mem q.2)
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
p : ↑s
q : ↑(s ∩ Ioi ↑p)
⊢ u' ↑p ∩ v ↑p ↑q ⊆ u ↑p ↑q ∩ v ↑p ↑q

tactic: congr
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
⊢ (∑' (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u ↑p ↑q ∩ v ↑p ↑q)) = ∑' (p : ↑s) (x : ↑(s ∩ Ioi ↑p)), 0

tactic: ext1 p
state_before: case e_f
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
⊢ (fun p => ∑' (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u ↑p ↑q ∩ v ↑p ↑q)) = fun p => ∑' (x : ↑(s ∩ Ioi ↑p)), 0

tactic: congr
state_before: case e_f.h
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
p : ↑s
⊢ (∑' (q : ↑(s ∩ Ioi ↑p)), ↑↑μ (u ↑p ↑q ∩ v ↑p ↑q)) = ∑' (x : ↑(s ∩ Ioi ↑p)), 0

tactic: ext1 q
state_before: case e_f.h.e_f
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
p : ↑s
⊢ (fun q => ↑↑μ (u ↑p ↑q ∩ v ↑p ↑q)) = fun x => 0

tactic: exact (huv p q).2.2.2.2 p.2 q.2.1 q.2.2
state_before: case e_f.h.e_f.h
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
p : ↑s
q : ↑(s ∩ Ioi ↑p)
⊢ ↑↑μ (u ↑p ↑q ∩ v ↑p ↑q) = 0

tactic: simp only [tsum_zero]
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
⊢ (∑' (p : ↑s) (x : ↑(s ∩ Ioi ↑p)), 0) = 0

tactic: have : ∀ᵐ x ∂μ, x ∉ t := by
  have : μ t = 0 := le_antisymm μt bot_le
  change μ _ = 0
  convert this
  ext y
  simp only [not_exists, exists_prop, mem_setOf_eq, mem_compl_iff, not_not_mem]
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
⊢ ∀ᵐ (x : α) ∂μ, f x = f' x

tactic: filter_upwards [this]with x hx
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
⊢ ∀ᵐ (x : α) ∂μ, f x = f' x

tactic: apply (iInf_eq_of_forall_ge_of_forall_gt_exists_lt _ _).symm
state_before: case h
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
⊢ f x = ⨅ (i : ↑s), piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x

tactic: have : μ t = 0 := le_antisymm μt bot_le
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
⊢ ∀ᵐ (x : α) ∂μ, ¬x ∈ t

tactic: change μ _ = 0
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ↑↑μ t = 0
⊢ ∀ᵐ (x : α) ∂μ, ¬x ∈ t

tactic: convert this
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ↑↑μ t = 0
⊢ ↑↑μ ({x | (fun x => ¬x ∈ t) x}ᶜ) = 0

tactic: ext y
state_before: case h.e'_2.h.e'_3
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ↑↑μ t = 0
⊢ {x | (fun x => ¬x ∈ t) x}ᶜ = t

tactic: simp only [not_exists, exists_prop, mem_setOf_eq, mem_compl_iff, not_not_mem]
state_before: case h.e'_2.h.e'_3.h
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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ↑↑μ t = 0
y : α
⊢ y ∈ {x | (fun x => ¬x ∈ t) x}ᶜ ↔ y ∈ t

tactic: intro i
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
⊢ ∀ (i : ↑s), f x ≤ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x

tactic: by_cases H : x ∈ u' i
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
i : ↑s
⊢ f x ≤ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x

tactic: swap
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
i : ↑s
H : x ∈ u' ↑i
⊢ f x ≤ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x

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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
i : ↑s
H : ¬x ∈ u' ↑i
⊢ f x ≤ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x

tactic: simp only [H, piecewise_eq_of_mem]
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
i : ↑s
H : x ∈ u' ↑i
⊢ f x ≤ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x

tactic: contrapose! hx
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
i : ↑s
H : x ∈ u' ↑i
⊢ f x ≤ ↑i

tactic: obtain ⟨r, ⟨xr, rq⟩, rs⟩ : ∃ r, r ∈ Ioo (i : β) (f x) ∩ s :=
  dense_iff_inter_open.1 s_dense (Ioo i (f x)) isOpen_Ioo (nonempty_Ioo.2 hx)
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
i : ↑s
H : x ∈ u' ↑i
hx : ↑i < f x
⊢ x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q

tactic: have A : x ∈ v i r := (huv i r).2.2.2.1 rq
state_before: case pos.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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
i : ↑s
H : x ∈ u' ↑i
hx : ↑i < f x
r : β
rs : r ∈ s
xr : ↑i < r
rq : r < f x
⊢ x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q

tactic: refine mem_iUnion.2 ⟨i, ?_⟩
state_before: case pos.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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
i : ↑s
H : x ∈ u' ↑i
hx : ↑i < f x
r : β
rs : r ∈ s
xr : ↑i < r
rq : r < f x
A : x ∈ v (↑i) r
⊢ x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q

tactic: refine mem_iUnion.2 ⟨⟨r, ⟨rs, xr⟩⟩, ?_⟩
state_before: case pos.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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
i : ↑s
H : x ∈ u' ↑i
hx : ↑i < f x
r : β
rs : r ∈ s
xr : ↑i < r
rq : r < f x
A : x ∈ v (↑i) r
⊢ x ∈ ⋃ (q : ↑(s ∩ Ioi ↑i)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) ∩ v ↑i ↑q

tactic: exact ⟨H, A⟩
state_before: case pos.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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
i : ↑s
H : x ∈ u' ↑i
hx : ↑i < f x
r : β
rs : r ∈ s
xr : ↑i < r
rq : r < f x
A : x ∈ v (↑i) r
⊢ x ∈ (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) ∩ v ↑i ↑{ val := r, property := (_ : r ∈ s ∧ r ∈ Ioi ↑i) }

tactic: simp only [H, le_top, not_false_iff, piecewise_eq_of_not_mem]
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
i : ↑s
H : ¬x ∈ u' ↑i
⊢ f x ≤ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x

tactic: intro q hq
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
⊢ ∀ (w : β), f x < w → ∃ i, piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x < w

tactic: obtain ⟨r, ⟨xr, rq⟩, rs⟩ : ∃ r, r ∈ Ioo (f x) q ∩ s :=
  dense_iff_inter_open.1 s_dense (Ioo (f x) q) isOpen_Ioo (nonempty_Ioo.2 hq)
state_before: 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
q : β
hq : f x < q
⊢ ∃ i, piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x < q

tactic: refine' ⟨⟨r, rs⟩, _⟩
state_before: case 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
q : β
hq : f x < q
r : β
rs : r ∈ s
xr : f x < r
rq : r < q
⊢ ∃ i, piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑i), u (↑i) q) (fun x => ↑i) (fun x => ⊤) x < q

tactic: have A : x ∈ u' r := mem_biInter fun i _ => (huv r i).2.2.1 xr
state_before: case 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
q : β
hq : f x < q
r : β
rs : r ∈ s
xr : f x < r
rq : r < q
⊢ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑{ val := r, property := rs }), u (↑{ val := r, property := rs }) q)
      (fun x => ↑{ val := r, property := rs }) (fun x => ⊤) x <
    q

tactic: simp only [A, rq, piecewise_eq_of_mem, Subtype.coe_mk]
state_before: case 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 : β),
    p ∈ s →
      ∀ (q : β),
        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 : β) (_ : q ∈ s ∩ Ioi p), u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ (i : ↑s), piecewise (u' ↑i) (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), u' ↑p ∩ v ↑p ↑q
μt : ↑↑μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, ¬x ∈ t
x : α
hx : ¬x ∈ ⋃ (p : ↑s) (q : ↑(s ∩ Ioi ↑p)), (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑p), u (↑p) q) ∩ v ↑p ↑q
q : β
hq : f x < q
r : β
rs : r ∈ s
xr : f x < r
rq : r < q
A : x ∈ u' r
⊢ piecewise (⋂ (q : β) (_ : q ∈ s ∩ Ioi ↑{ val := r, property := rs }), u (↑{ val := r, property := rs }) q)
      (fun x => ↑{ val := r, property := rs }) (fun x => ⊤) x <
    q