awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_3 : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs :
  ∀ (n : ℕ),
    ∃ r ∈ Ioo 0 (Rs n), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)
refine_1 :
  Tendsto (β := ℝ)
    (fun (n : ℕ) ↦
      ⋯.choose (p := fun (r : ℝ) ↦
        r ∈ Ioo 0 (Rs n) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
    atTop (𝓝 0)
refine_2 :
  ∀ (n : ℕ),
    LT.lt (α := ℝ) 0
        ((fun (n : ℕ) ↦
            ⋯.choose (p := fun (r : ℝ) ↦
              r ∈ Ioo 0 (Rs n) ∧
                Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
          n) ∧
      Eq (α := ℝ≥0∞)
        ((μ : (a : Set Ω) → ℝ≥0∞)
            (frontier
              (Metric.thickening
                ((fun (n : ℕ) ↦
                    ⋯.choose (p := fun (r : ℝ) ↦
                      r ∈ Ioo 0 (Rs n) ∧
                        Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
                  n)
                s)) :
          ℝ≥0∞)
        (0 : ℝ≥0∞)
⊢ ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) s)) : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
refine ⟨fun n : ℕ => (obs n).choose, ⟨?_, ?_⟩⟩
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_3 : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs :
  ∀ (n : ℕ),
    ∃ r ∈ Ioo 0 (Rs n), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Tendsto (β := ℝ)
    (fun (n : ℕ) ↦
      ⋯.choose (p := fun (r : ℝ) ↦
        r ∈ Ioo 0 (Rs n) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
    atTop (𝓝 0)

Tactic:
tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds Rs_lim
      (fun n ↦ (obs n).choose_spec.1.1.le) fun n ↦ (obs n).choose_spec.1.2.le

Premises used: Exists.choose_spec, tendsto_of_tendsto_of_tendsto_of_le_of_le, tendsto_const_nhds, LT.lt.le, And.right, And.left
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_3 : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs :
  ∀ (n : ℕ),
    ∃ r ∈ Ioo 0 (Rs n), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Tendsto (β := ℝ)
    (fun (n : ℕ) ↦
      ⋯.choose (p := fun (r : ℝ) ↦
        r ∈ Ioo 0 (Rs n) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
    atTop (𝓝 0)

Tactic:
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds Rs_lim
      (fun n ↦ (obs n).choose_spec.1.1.le) fun n ↦ (obs n).choose_spec.1.2.le

Premises used: Exists.choose_spec, tendsto_of_tendsto_of_tendsto_of_le_of_le, tendsto_const_nhds, LT.lt.le, And.right, And.left
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_3 : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs :
  ∀ (n : ℕ),
    ∃ r ∈ Ioo 0 (Rs n), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ∀ (n : ℕ),
    LT.lt (α := ℝ) 0
        ((fun (n : ℕ) ↦
            ⋯.choose (p := fun (r : ℝ) ↦
              r ∈ Ioo 0 (Rs n) ∧
                Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
          n) ∧
      Eq (α := ℝ≥0∞)
        ((μ : (a : Set Ω) → ℝ≥0∞)
            (frontier
              (Metric.thickening
                ((fun (n : ℕ) ↦
                    ⋯.choose (p := fun (r : ℝ) ↦
                      r ∈ Ioo 0 (Rs n) ∧
                        Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
                  n)
                s)) :
          ℝ≥0∞)
        (0 : ℝ≥0∞)

Tactic:
fun n ↦ ⟨(obs n).choose_spec.1.1, (obs n).choose_spec.2⟩

Premises used: Exists.choose_spec, And.right, And.left, And.intro
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_3 : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs :
  ∀ (n : ℕ),
    ∃ r ∈ Ioo 0 (Rs n), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ∀ (n : ℕ),
    LT.lt (α := ℝ) 0
        ((fun (n : ℕ) ↦
            ⋯.choose (p := fun (r : ℝ) ↦
              r ∈ Ioo 0 (Rs n) ∧
                Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
          n) ∧
      Eq (α := ℝ≥0∞)
        ((μ : (a : Set Ω) → ℝ≥0∞)
            (frontier
              (Metric.thickening
                ((fun (n : ℕ) ↦
                    ⋯.choose (p := fun (r : ℝ) ↦
                      r ∈ Ioo 0 (Rs n) ∧
                        Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening r s)) : ℝ≥0∞) (0 : ℝ≥0∞)))
                  n)
                s)) :
          ℝ≥0∞)
        (0 : ℝ≥0∞)

Tactic:
exact fun n ↦ ⟨(obs n).choose_spec.1.1, (obs n).choose_spec.2⟩
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
inl :
  ∀
    (h :
      ∀ {E : Set Ω},
        MeasurableSet E →
          Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
            Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) ⊥
              (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)),
    LE.le (α := ℝ≥0∞) (limsup (fun (i : ι) ↦ (μs i : (a : Set Ω) → ℝ≥0∞) F) ⊥ : ℝ≥0∞)
      ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞)
inr :
  L.NeBot →
    LE.le (α := ℝ≥0∞) (limsup (fun (i : ι) ↦ (μs i : (a : Set Ω) → ℝ≥0∞) F) L : ℝ≥0∞)
      ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (limsup (fun (i : ι) ↦ (μs i : (a : Set Ω) → ℝ≥0∞) F) L : ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞)

Tactic:
rcases L.eq_or_neBot with rfl | _

Premises used: Filter.eq_or_neBot, rfl
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) ⊥
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (limsup (fun (i : ι) ↦ (μs i : (a : Set Ω) → ℝ≥0∞) F) ⊥ : ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞)

Tactic:
simp only [limsup_bot, bot_eq_zero', zero_le]

Premises used: zero_le, Filter.limsup_bot, bot_eq_zero'
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
⊢ ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
have ex := exists_null_frontiers_thickening μ F
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
⊢ Tendsto rs atTop (𝓝 0)

Tactic:
have rs_lim : Tendsto rs atTop (𝓝 0) := (Classical.choose_spec ex).1
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
⊢ ∀ (n : ℕ), 0 < rs n

Tactic:
have rs_pos : ∀ n, 0 < rs n := fun n ↦ ((Classical.choose_spec ex).2 n).1
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
⊢ ∀ (n : ℕ), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
have rs_null : ∀ n, μ (frontier (Metric.thickening (rs n) F)) = 0 :=
    fun n ↦ ((Classical.choose_spec ex).2 n).2
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ∀ (n : ℕ), IsOpen[inst_1] (Metric.thickening (rs n) F)

Tactic:
have Fthicks_open : ∀ n, IsOpen (Metric.thickening (rs n) F) :=
    fun n ↦ Metric.isOpen_thickening
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
Fthicks_open : ∀ (n : ℕ), IsOpen[inst_1] (Metric.thickening (rs n) F)
⊢ ∀ (n : ℕ),
    Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F) : ℝ≥0∞)) L
      (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F)) : Filter ℝ≥0∞)

Tactic:
have key := fun (n : ℕ) ↦ h (Fthicks_open n).measurableSet (rs_null n)
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
Fthicks_open : ∀ (n : ℕ), IsOpen[inst_1] (Metric.thickening (rs n) F)
key :
  ∀ (n : ℕ),
    Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F) : ℝ≥0∞)) L
      (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F)) : Filter ℝ≥0∞)
inr :
  ∀ (ε : ℝ≥0),
    0 < ε →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞) (∞ : ℝ≥0∞) →
        limsup (fun (i : ι) ↦ (μs i : (a : Set Ω) → ℝ≥0∞) F) L ≤
          HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞) (↑ε : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (limsup (fun (i : ι) ↦ (μs i : (a : Set Ω) → ℝ≥0∞) F) L : ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞)

Tactic:
apply ENNReal.le_of_forall_pos_le_add

Premises used: ENNReal.le_of_forall_pos_le_add
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
Fthicks_open : ∀ (n : ℕ), IsOpen[inst_1] (Metric.thickening (rs n) F)
key :
  ∀ (n : ℕ),
    Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F) : ℝ≥0∞)) L
      (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F)) : Filter ℝ≥0∞)
ε : ℝ≥0
ε_pos : 0 < ε
μF_finite : LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ GT.gt (α := ℝ) 1 0

Tactic:
simp only [gt_iff_lt, zero_lt_one]

Premises used: zero_lt_one, gt_iff_lt
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
Fthicks_open : ∀ (n : ℕ), IsOpen[inst_1] (Metric.thickening (rs n) F)
key :
  ∀ (n : ℕ),
    Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F) : ℝ≥0∞)) L
      (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F)) : Filter ℝ≥0∞)
ε : ℝ≥0
ε_pos : 0 < ε
μF_finite : LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Tendsto (β := ℝ≥0∞) (fun (r : ℝ) ↦ ((μ : (a : Set Ω) → ℝ≥0∞) (Metric.cthickening r F) : ℝ≥0∞)) (𝓝 0)
    (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) F) : Filter ℝ≥0∞)

Tactic:
have keyB := tendsto_measure_cthickening_of_isClosed (μ := μ) (s := F)
                ⟨1, ⟨by simp only [gt_iff_lt, zero_lt_one], measure_ne_top _ _⟩⟩ F_closed
-- Module: Mathlib.MeasureTheory.Measure.Portmanteau
Goal:
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst : MeasurableSpace Ω
inst_1 : TopologicalSpace Ω
inst_2 : PseudoMetrizableSpace Ω
inst_3 : OpensMeasurableSpace Ω
μ : Measure Ω
inst_4 : IsFiniteMeasure μ
μs : ι → Measure Ω
h :
  ∀ {E : Set Ω},
    MeasurableSet E →
      Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier E) : ℝ≥0∞) (0 : ℝ≥0∞) →
        Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) E : ℝ≥0∞)) L
          (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) E) : Filter ℝ≥0∞)
F : Set Ω
F_closed : IsClosed[inst_1] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h_1 : L.NeBot
ex :
  ∃ (rs : ℕ → ℝ),
    Tendsto rs atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < rs n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
rs : ℕ → ℝ :=
  Classical.choose (p := fun (x : ℕ → ℝ) ↦
    Tendsto x atTop (𝓝 0) ∧
      ∀ (n : ℕ),
        0 < x n ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (x n) F)) : ℝ≥0∞) (0 : ℝ≥0∞))
    ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), Eq (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (frontier (Metric.thickening (rs n) F)) : ℝ≥0∞) (0 : ℝ≥0∞)
Fthicks_open : ∀ (n : ℕ), IsOpen[inst_1] (Metric.thickening (rs n) F)
key :
  ∀ (n : ℕ),
    Tendsto (β := ℝ≥0∞) (fun (i : ι) ↦ ((μs i : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F) : ℝ≥0∞)) L
      (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) (Metric.thickening (rs n) F)) : Filter ℝ≥0∞)
ε : ℝ≥0
ε_pos : 0 < ε
μF_finite : LT.lt (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞) (∞ : ℝ≥0∞)
keyB :
  Tendsto (β := ℝ≥0∞) (fun (r : ℝ) ↦ ((μ : (a : Set Ω) → ℝ≥0∞) (Metric.cthickening r F) : ℝ≥0∞)) (𝓝 0)
    (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) F) : Filter ℝ≥0∞)
⊢ Membership.mem (γ := Filter ℝ≥0∞) (𝓝 ((μ : (a : Set Ω) → ℝ≥0∞) F) : Filter ℝ≥0∞)
    (Iio (HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) F : ℝ≥0∞) (↑ε : ℝ≥0∞)))

Tactic:
have nhds : Iio (μ F + ε) ∈ 𝓝 (μ F) :=
    Iio_mem_nhds <| ENNReal.lt_add_right μF_finite.ne (ENNReal.coe_pos.mpr ε_pos).ne'