l3lab/ntp-mathlib huggingface dataset:
theorem isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace [TopologicalSpace α]
    [MeasurableSpace α] {μ : Measure α} [CompactSpace α] :
    IsFiniteMeasure μ ↔ IsFiniteMeasureOnCompacts μ := by
  constructor <;> 

Proof state:
case mpr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : CompactSpace α
⊢ IsFiniteMeasureOnCompacts μ → IsFiniteMeasure μ

Next tactic:
intros
theorem isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace [TopologicalSpace α]
    [MeasurableSpace α] {μ : Measure α} [CompactSpace α] :
    IsFiniteMeasure μ ↔ IsFiniteMeasureOnCompacts μ := by
  constructor <;> intros
  · 

Proof state:
case mp
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : CompactSpace α
a✝ : IsFiniteMeasure μ
⊢ IsFiniteMeasureOnCompacts μ

Next tactic:
infer_instance
theorem isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace [TopologicalSpace α]
    [MeasurableSpace α] {μ : Measure α} [CompactSpace α] :
    IsFiniteMeasure μ ↔ IsFiniteMeasureOnCompacts μ := by
  constructor <;> intros
  · infer_instance
  · 

Proof state:
case mpr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : CompactSpace α
a✝ : IsFiniteMeasureOnCompacts μ
⊢ IsFiniteMeasure μ

Next tactic:
exact CompactSpace.isFiniteMeasure
/-- 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
  

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
⊢ FiniteSpanningSetsIn μ {K | IsOpen K}

Next tactic:
suffices H : Nonempty (μ.FiniteSpanningSetsIn { K | IsOpen K })
/-- 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 })
  

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 (FiniteSpanningSetsIn μ {K | IsOpen K})
⊢ FiniteSpanningSetsIn μ {K | IsOpen K}
case H
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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
exact H.some
/-- 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
  

Proof state:
case H
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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
cases isEmpty_or_nonempty α
/-- 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
  · 

Proof state:
case H.inl
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✝ : IsEmpty α
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
exact
      ⟨{  set := fun _ => ∅
          set_mem := fun _ => by simp
          finite := fun _ => by simp
          spanning := by simp [eq_iff_true_of_subsingleton] }⟩
/-- 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 

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✝ : IsEmpty α
x✝ : ℕ
⊢ (fun x => ∅) x✝ ∈ {K | IsOpen K}

Next tactic:
simp
/-- 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 

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✝ : IsEmpty α
x✝ : ℕ
⊢ ↑↑μ ((fun x => ∅) x✝) < ⊤

Next tactic:
simp
/-- 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 

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✝ : IsEmpty α
⊢ ⋃ i, (fun x => ∅) i = univ

Next tactic:
simp [eq_iff_true_of_subsingleton]
/-- 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] }⟩
  

Proof state:
case H.inr
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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
inhabit α
/-- 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
  

Proof state:
case H.inr
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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ }
/-- 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 < ∞ }
  

Proof state:
case H.inr
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 < ⊤}
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
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
/-- 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
  

Proof state:
case H.inr.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 = ⋃₀ S
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
rw [μ.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT
/-- 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
  

Proof state:
case H.inr.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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
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)
/-- 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
    

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
⊢ Set.Nonempty T

Next tactic:
by_contra h'T
/-- 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
    

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
h'T : ¬Set.Nonempty T
⊢ False

Next tactic:
rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT
/-- 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
    

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 : ∅ = univ
h'T : ¬Set.Nonempty T
⊢ False

Next tactic:
simpa only [← hT] using mem_univ (default : α)
/-- 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)
  

Proof state:
case H.inr.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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, T = range f
/-- 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
  

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, T = range f
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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
exact T_count.exists_eq_range T_ne
/-- 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
  

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
⊢ Nonempty (FiniteSpanningSetsIn μ {K | IsOpen K})

Next tactic:
have fS : ∀ n, f n ∈ S := by
    intro n
    apply TS
    rw [hf]
    exact mem_range_self n