l3lab/ntp-mathlib huggingface dataset:
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ u n ∈ T'

Next tactic:
rw [hu]
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ u n ∈ range u

Next tactic:
exact mem_range_self n
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    

Proof state:
case intro.intro.intro.intro.intro.refine'_1
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
⊢ IsOpen (u n)

Next tactic:
rcases mem_insert_iff.1 this with h|h
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.inl
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen (u n)

Next tactic:
rw [h]
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.inl
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen ∅

Next tactic:
exact isOpen_empty
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.inr
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n ∈ T
⊢ IsOpen (u n)

Next tactic:
exact T_open _ h
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · 

Proof state:
case intro.intro.intro.intro.intro.refine'_2
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
⊢ s ⊆ ⋃ n, u n

Next tactic:
intro x hx
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · intro x hx
    

Proof state:
case intro.intro.intro.intro.intro.refine'_2
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
⊢ x ∈ ⋃ n, u n

Next tactic:
obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
⊢ ∃ v ∈ T, x ∈ v

Next tactic:
simpa only [mem_iUnion, exists_prop] using sT hx
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
    

Proof state:
case intro.intro.intro.intro.intro.refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ x ∈ ⋃ n, u n

Next tactic:
have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
    have : v ∈ range u := by 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ v ∈ range u

Next tactic:
rw [← hu]
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
    have : v ∈ range u := by rw [← hu]; 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ v ∈ T'

Next tactic:
exact subset_insert ∅ T hv
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
    have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv
    

Proof state:
case intro.intro.intro.intro.intro.refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
this : v ∈ range u
⊢ x ∈ ⋃ n, u n

Next tactic:
obtain ⟨n, rfl⟩ : ∃ n, u n = v := by simpa only [mem_range] using this
/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
  let T' : Set (Set X) := insert ∅ T
  have T'_count : T'.Countable := Countable.insert ∅ T_count
  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
  refine' ⟨u, _, _, _⟩
  · intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h|h
    · rw [h]
      exact isOpen_empty
    · exact T_open _ h
  · intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
    have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv
    obtain ⟨n, rfl⟩ : ∃ n, u n = v := by 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
this : v ∈ range u
⊢ ∃ n, u n = v

Next tactic:
simpa only [mem_range] using this