l3lab/ntp-mathlib huggingface dataset:
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    

Proof state:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
⊢ ∀ (n : ℕ), f n ∈ S

Next tactic:
intro n
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    

Proof state:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ S

Next tactic:
apply TS
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    

Proof state:
case a
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ T

Next tactic:
rw [hf]
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    

Proof state:
case a
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ range f

Next tactic:
exact mem_range_self n
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  

Proof state:
case H.inr.intro.intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
  

Proof state:
case H.inr.intro.intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
⊢ ⋃ i, f i = univ

Next tactic:
refine eq_univ_of_forall fun x => ?_
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
  refine eq_univ_of_forall fun x => ?_
  

Proof state:
case H.inr.intro.intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
x : α
⊢ x ∈ ⋃ i, f i

Next tactic:
obtain ⟨t, tT, xt⟩ : ∃ t : Set α, t ∈ range f ∧ x ∈ t := by
    have : x ∈ ⋃₀ T := by simp only [hT, mem_univ]
    simpa only [mem_sUnion, exists_prop, ← hf]
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
  refine eq_univ_of_forall fun x => ?_
  obtain ⟨t, tT, xt⟩ : ∃ t : Set α, t ∈ range f ∧ x ∈ t := by
    

Proof state:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
x : α
⊢ ∃ t ∈ range f, x ∈ t

Next tactic:
have : x ∈ ⋃₀ T := by simp only [hT, mem_univ]
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
  refine eq_univ_of_forall fun x => ?_
  obtain ⟨t, tT, xt⟩ : ∃ t : Set α, t ∈ range f ∧ x ∈ t := by
    have : x ∈ ⋃₀ T := by 

Proof state:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
x : α
⊢ x ∈ ⋃₀ T

Next tactic:
simp only [hT, mem_univ]
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
  refine eq_univ_of_forall fun x => ?_
  obtain ⟨t, tT, xt⟩ : ∃ t : Set α, t ∈ range f ∧ x ∈ t := by
    have : x ∈ ⋃₀ T := by simp only [hT, mem_univ]
    

Proof state:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
x : α
this : x ∈ ⋃₀ T
⊢ ∃ t ∈ range f, x ∈ t

Next tactic:
simpa only [mem_sUnion, exists_prop, ← hf]
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
  refine eq_univ_of_forall fun x => ?_
  obtain ⟨t, tT, xt⟩ : ∃ t : Set α, t ∈ range f ∧ x ∈ t := by
    have : x ∈ ⋃₀ T := by simp only [hT, mem_univ]
    simpa only [mem_sUnion, exists_prop, ← hf]
  

Proof state:
case H.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
x : α
t : Set α
tT : t ∈ range f
xt : x ∈ t
⊢ x ∈ ⋃ i, f i

Next tactic:
obtain ⟨n, rfl⟩ : ∃ n : ℕ, f n = t := by simpa only using tT
/-- A locally finite measure on a second countable topological space admits a finite spanning
sequence of open sets. -/
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } := by
  suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
  exact H.some
  cases isEmpty_or_nonempty α
  · exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
  inhabit α
  let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
  obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
    isOpen_sUnion_countable S fun s hs => hs.1
  rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
  have T_ne : T.Nonempty := by
    by_contra h'T
    rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
    simpa only [← hT] using mem_univ (default : α)
  obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
  exact T_count.exists_eq_range T_ne
  have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n
  refine'
    ⟨{  set := f
        set_mem := fun n => (fS n).1
        finite := fun n => (fS n).2
        spanning := _ }⟩
  refine eq_univ_of_forall fun x => ?_
  obtain ⟨t, tT, xt⟩ : ∃ t : Set α, t ∈ range f ∧ x ∈ t := by
    have : x ∈ ⋃₀ T := by simp only [hT, mem_univ]
    simpa only [mem_sUnion, exists_prop, ← hf]
  obtain ⟨n, rfl⟩ : ∃ n : ℕ, f n = t := by 

Proof state:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen s ∧ ↑↑μ s < ⊤}
T : Set (Set α)
T_count : Set.Countable T
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : Set.Nonempty T
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
x : α
t : Set α
tT : t ∈ range f
xt : x ∈ t
⊢ ∃ n, f n = t

Next tactic:
simpa only using tT