l3lab/ntp-mathlib huggingface dataset:
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs✝ : ¬ms = ⊤
r : ℝ≥0∞
hs : r > ms
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r

Next tactic:
have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by 

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs✝ : ¬ms = ⊤
r : ℝ≥0∞
hs : r > ms
⊢ ∃ t, MeasurableSet t ∧ s ⊆ t ∧ ↑m t < r

Next tactic:
simpa [iInf_lt_iff] using hs
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs✝ : ¬ms = ⊤
r : ℝ≥0∞
hs : r > ms
this : ∃ t, MeasurableSet t ∧ s ⊆ t ∧ ↑m t < r
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r

Next tactic:
rcases this with ⟨t, hmt, hin, hlt⟩
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs✝ : ¬ms = ⊤
r : ℝ≥0∞
hs : r > ms
t : Set α
hmt : MeasurableSet t
hin : s ⊆ t
hlt : ↑m t < r
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r

Next tactic:
exists t
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms

Next tactic:
have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
⊢ ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹

Next tactic:
intro n
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
n : ℕ
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹

Next tactic:
refine' this _ (ENNReal.lt_add_right hs _)
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
n : ℕ
⊢ (↑n)⁻¹ ≠ 0

Next tactic:
simp
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
this : ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms

Next tactic:
choose t hsub hm hm' using this
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms

Next tactic:
refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
    

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
⊢ ↑m (⋂ n, t n) = ms

Next tactic:
have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
      tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
    have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
      tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero
    

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 (ms + 0))
⊢ ↑m (⋂ n, t n) = ms

Next tactic:
rw [add_zero] at this
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
    have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
      tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero
    rw [add_zero] at this
    

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)
⊢ ↑m (⋂ n, t n) = ms

Next tactic:
refine' le_antisymm (ge_of_tendsto' this fun n => _) _
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
    have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
      tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero
    rw [add_zero] at this
    refine' le_antisymm (ge_of_tendsto' this fun n => _) _
    · 

Proof state:
case neg.refine'_1
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)
n : ℕ
⊢ ↑m (⋂ n, t n) ≤ ms + (↑n)⁻¹

Next tactic:
exact le_trans (m.mono' <| iInter_subset t n) (hm' n).le
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
    have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
      tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero
    rw [add_zero] at this
    refine' le_antisymm (ge_of_tendsto' this fun n => _) _
    · exact le_trans (m.mono' <| iInter_subset t n) (hm' n).le
    · 

Proof state:
case neg.refine'_2
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)
⊢ ms ≤ ↑m (⋂ n, t n)

Next tactic:
refine' iInf_le_of_le (⋂ n, t n) _
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
    have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
      tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero
    rw [add_zero] at this
    refine' le_antisymm (ge_of_tendsto' this fun n => _) _
    · exact le_trans (m.mono' <| iInter_subset t n) (hm' n).le
    · refine' iInf_le_of_le (⋂ n, t n) _
      

Proof state:
case neg.refine'_2
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)
⊢ ⨅ (_ : s ⊆ ⋂ n, t n), ⨅ (_ : MeasurableSet (⋂ n, t n)), ↑m (⋂ n, t n) ≤ ↑m (⋂ n, t n)

Next tactic:
refine' iInf_le_of_le (subset_iInter hsub) _
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s := by
  simp only [trim_eq_iInf]; set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t
  by_cases hs : ms = ∞
  · simp only [hs]
    simp only [iInf_eq_top] at hs
    exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩
  · have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
      intro r hs
      have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
      rcases this with ⟨t, hmt, hin, hlt⟩
      exists t
    have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
      intro n
      refine' this _ (ENNReal.lt_add_right hs _)
      simp
    choose t hsub hm hm' using this
    refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩
    have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
      tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero
    rw [add_zero] at this
    refine' le_antisymm (ge_of_tendsto' this fun n => _) _
    · exact le_trans (m.mono' <| iInter_subset t n) (hm' n).le
    · refine' iInf_le_of_le (⋂ n, t n) _
      refine' iInf_le_of_le (subset_iInter hsub) _
      

Proof state:
case neg.refine'_2
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹
this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)
⊢ ⨅ (_ : MeasurableSet (⋂ n, t n)), ↑m (⋂ n, t n) ≤ ↑m (⋂ n, t n)

Next tactic:
refine' iInf_le _ (MeasurableSet.iInter hm)
theorem exists_measurable_superset_of_trim_eq_zero {m : OuterMeasure α} {s : Set α}
    (h : m.trim s = 0) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = 0 := by
  

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
h : ↑(trim m) s = 0
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = 0

Next tactic:
rcases exists_measurable_superset_eq_trim m s with ⟨t, hst, ht, hm⟩
theorem exists_measurable_superset_of_trim_eq_zero {m : OuterMeasure α} {s : Set α}
    (h : m.trim s = 0) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = 0 := by
  rcases exists_measurable_superset_eq_trim m s with ⟨t, hst, ht, hm⟩
  

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
h : ↑(trim m) s = 0
t : Set α
hst : s ⊆ t
ht : MeasurableSet t
hm : ↑m t = ↑(trim m) s
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = 0

Next tactic:
exact ⟨t, hst, ht, h ▸ hm⟩
/-- If `μ i` is a countable family of outer measures, then for every set `s` there exists
a measurable set `t ⊇ s` such that `μ i t = (μ i).trim s` for all `i`. -/
theorem exists_measurable_superset_forall_eq_trim {ι} [Countable ι] (μ : ι → OuterMeasure α)
    (s : Set α) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ i, μ i t = (μ i).trim s := by
  

Proof state:
α : Type u_1
inst✝¹ : MeasurableSpace α
m : OuterMeasure α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
s : Set α
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s

Next tactic:
choose t hst ht hμt using fun i => (μ i).exists_measurable_superset_eq_trim s
/-- If `μ i` is a countable family of outer measures, then for every set `s` there exists
a measurable set `t ⊇ s` such that `μ i t = (μ i).trim s` for all `i`. -/
theorem exists_measurable_superset_forall_eq_trim {ι} [Countable ι] (μ : ι → OuterMeasure α)
    (s : Set α) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ i, μ i t = (μ i).trim s := by
  choose t hst ht hμt using fun i => (μ i).exists_measurable_superset_eq_trim s
  

Proof state:
α : Type u_1
inst✝¹ : MeasurableSpace α
m : OuterMeasure α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
s : Set α
t : ι → Set α
hst : ∀ (i : ι), s ⊆ t i
ht : ∀ (i : ι), MeasurableSet (t i)
hμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s

Next tactic:
replace hst := subset_iInter hst