l3lab/ntp-mathlib huggingface dataset:
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ↑↑(Content.measure μ) A < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ↑↑(Content.measure μ) U < r

Next tactic:
rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ⨅ U, ⨅ (hU : IsOpen U), ⨅ (_ : A ⊆ U), innerContent μ { carrier := U, is_open' := hU } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ↑↑(Content.measure μ) U < r

Next tactic:
simp only [iInf_lt_iff] at hr
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ∃ i, ∃ (h : IsOpen i) (_ : A ⊆ i), innerContent μ { carrier := i, is_open' := (_ : IsOpen i) } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ↑↑(Content.measure μ) U < r

Next tactic:
rcases hr with ⟨U, hUo, hAU, hr⟩
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    

Proof state:
case intro.intro.intro
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
U : Set G
hUo : IsOpen U
hAU : A ⊆ U
hr : innerContent μ { carrier := U, is_open' := (_ : IsOpen U) } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ↑↑(Content.measure μ) U < r

Next tactic:
rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    

Proof state:
case intro.intro.intro
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
U : Set G
hUo : IsOpen U
hAU : A ⊆ U
hr : ↑↑(Content.measure μ) U < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ↑↑(Content.measure μ) U < r

Next tactic:
exact ⟨U, hAU, hUo, hr⟩
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : Measure.OuterRegular (Content.measure μ)
⊢ Measure.Regular (Content.measure μ)

Next tactic:
have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : Measure.OuterRegular (Content.measure μ)
⊢ IsFiniteMeasureOnCompacts (Content.measure μ)

Next tactic:
refine' ⟨fun K hK => _⟩
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : Measure.OuterRegular (Content.measure μ)
K : Set G
hK : IsCompact K
⊢ ↑↑(Content.measure μ) K < ⊤

Next tactic:
rw [measure_apply _ hK.measurableSet]
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : Measure.OuterRegular (Content.measure μ)
K : Set G
hK : IsCompact K
⊢ ↑(Content.outerMeasure μ) K < ⊤

Next tactic:
exact μ.outerMeasure_lt_top_of_isCompact hK
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK
  

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this✝ : Measure.OuterRegular (Content.measure μ)
this : IsFiniteMeasureOnCompacts (Content.measure μ)
⊢ Measure.Regular (Content.measure μ)

Next tactic:
refine' ⟨fun U hU r hr => _⟩
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK
  refine' ⟨fun U hU r hr => _⟩
  

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this✝ : Measure.OuterRegular (Content.measure μ)
this : IsFiniteMeasureOnCompacts (Content.measure μ)
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : r < ↑↑(Content.measure μ) U
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ↑↑(Content.measure μ) K

Next tactic:
rw [measure_apply _ hU.measurableSet, μ.outerMeasure_of_isOpen U hU] at hr
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK
  refine' ⟨fun U hU r hr => _⟩
  rw [measure_apply _ hU.measurableSet, μ.outerMeasure_of_isOpen U hU] at hr
  

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this✝ : Measure.OuterRegular (Content.measure μ)
this : IsFiniteMeasureOnCompacts (Content.measure μ)
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : r < innerContent μ { carrier := U, is_open' := hU }
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ↑↑(Content.measure μ) K

Next tactic:
simp only [innerContent, lt_iSup_iff] at hr
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK
  refine' ⟨fun U hU r hr => _⟩
  rw [measure_apply _ hU.measurableSet, μ.outerMeasure_of_isOpen U hU] at hr
  simp only [innerContent, lt_iSup_iff] at hr
  

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this✝ : Measure.OuterRegular (Content.measure μ)
this : IsFiniteMeasureOnCompacts (Content.measure μ)
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : ∃ i, ∃ (_ : ↑i ⊆ ↑{ carrier := U, is_open' := hU }), r < ↑(toFun μ i)
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ↑↑(Content.measure μ) K

Next tactic:
rcases hr with ⟨K, hKU, hr⟩
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK
  refine' ⟨fun U hU r hr => _⟩
  rw [measure_apply _ hU.measurableSet, μ.outerMeasure_of_isOpen U hU] at hr
  simp only [innerContent, lt_iSup_iff] at hr
  rcases hr with ⟨K, hKU, hr⟩
  

Proof state:
case intro.intro
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this✝ : Measure.OuterRegular (Content.measure μ)
this : IsFiniteMeasureOnCompacts (Content.measure μ)
U : Set G
hU : IsOpen U
r : ℝ≥0∞
K : Compacts G
hKU : ↑K ⊆ ↑{ carrier := U, is_open' := hU }
hr : r < ↑(toFun μ K)
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ↑↑(Content.measure μ) K

Next tactic:
refine' ⟨K, hKU, K.2, hr.trans_le _⟩
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := by
  have : μ.measure.OuterRegular := by
    refine' ⟨fun A hA r (hr : _ < _) => _⟩
    rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
    simp only [iInf_lt_iff] at hr
    rcases hr with ⟨U, hUo, hAU, hr⟩
    rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
    exact ⟨U, hAU, hUo, hr⟩
  have : IsFiniteMeasureOnCompacts μ.measure := by
    refine' ⟨fun K hK => _⟩
    rw [measure_apply _ hK.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK
  refine' ⟨fun U hU r hr => _⟩
  rw [measure_apply _ hU.measurableSet, μ.outerMeasure_of_isOpen U hU] at hr
  simp only [innerContent, lt_iSup_iff] at hr
  rcases hr with ⟨K, hKU, hr⟩
  refine' ⟨K, hKU, K.2, hr.trans_le _⟩
  

Proof state:
case intro.intro
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : T2Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this✝ : Measure.OuterRegular (Content.measure μ)
this : IsFiniteMeasureOnCompacts (Content.measure μ)
U : Set G
hU : IsOpen U
r : ℝ≥0∞
K : Compacts G
hKU : ↑K ⊆ ↑{ carrier := U, is_open' := hU }
hr : r < ↑(toFun μ K)
⊢ ↑(toFun μ K) ≤ ↑↑(Content.measure μ) ↑K

Next tactic:
exact (μ.le_outerMeasure_compacts K).trans (le_toMeasure_apply _ _ _)
theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
    {ε : NNReal} (hε : ε ≠ 0) :
    ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε := by
  

Proof state:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ K', K.carrier ⊆ interior K'.carrier ∧ (fun s => ↑(toFun μ s)) K' ≤ (fun s => ↑(toFun μ s)) K + ↑ε

Next tactic:
by_contra hc
theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
    {ε : NNReal} (hε : ε ≠ 0) :
    ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε := by
  by_contra hc
  

Proof state:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc : ¬∃ K', K.carrier ⊆ interior K'.carrier ∧ (fun s => ↑(toFun μ s)) K' ≤ (fun s => ↑(toFun μ s)) K + ↑ε
⊢ False

Next tactic:
simp only [not_exists, not_and, not_le] at hc
theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
    {ε : NNReal} (hε : ε ≠ 0) :
    ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε := by
  by_contra hc
  simp only [not_exists, not_and, not_le] at hc
  

Proof state:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc : ∀ (x : Compacts G), K.carrier ⊆ interior x.carrier → ↑(toFun μ K) + ↑ε < ↑(toFun μ x)
⊢ False

Next tactic:
have lower_bound_iInf : μ K + ε ≤
      ⨅ (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
    le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
    {ε : NNReal} (hε : ε ≠ 0) :
    ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε := by
  by_contra hc
  simp only [not_exists, not_and, not_le] at hc
  have lower_bound_iInf : μ K + ε ≤
      ⨅ (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
    le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
  

Proof state:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc : ∀ (x : Compacts G), K.carrier ⊆ interior x.carrier → ↑(toFun μ K) + ↑ε < ↑(toFun μ x)
lower_bound_iInf : (fun s => ↑(toFun μ s)) K + ↑ε ≤ ⨅ K', ⨅ (_ : ↑K ⊆ interior ↑K'), (fun s => ↑(toFun μ s)) K'
⊢ False

Next tactic:
rw [← H] at lower_bound_iInf
theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
    {ε : NNReal} (hε : ε ≠ 0) :
    ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε := by
  by_contra hc
  simp only [not_exists, not_and, not_le] at hc
  have lower_bound_iInf : μ K + ε ≤
      ⨅ (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
    le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
  rw [← H] at lower_bound_iInf
  

Proof state:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc : ∀ (x : Compacts G), K.carrier ⊆ interior x.carrier → ↑(toFun μ K) + ↑ε < ↑(toFun μ x)
lower_bound_iInf : (fun s => ↑(toFun μ s)) K + ↑ε ≤ (fun s => ↑(toFun μ s)) K
⊢ False

Next tactic:
exact (lt_self_iff_false (μ K)).mp (lt_of_le_of_lt' lower_bound_iInf
    (ENNReal.lt_add_right (ne_top_of_lt (μ.lt_top K)) (ENNReal.coe_ne_zero.mpr hε)))
/-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
  on compact sets. -/
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K := by
  

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : T2Space G
inst✝ : BorelSpace G
H : ContentRegular μ
K : Compacts G
⊢ ↑↑(Content.measure μ) ↑K = (fun s => ↑(toFun μ s)) K

Next tactic:
refine' le_antisymm _ _
/-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
  on compact sets. -/
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K := by
  refine' le_antisymm _ _
  · 

Proof state:
case refine'_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : T2Space G
inst✝ : BorelSpace G
H : ContentRegular μ
K : Compacts G
⊢ ↑↑(Content.measure μ) ↑K ≤ (fun s => ↑(toFun μ s)) K

Next tactic:
apply ENNReal.le_of_forall_pos_le_add
/-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
  on compact sets. -/
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K := by
  refine' le_antisymm _ _
  · apply ENNReal.le_of_forall_pos_le_add
    

Proof state:
case refine'_1.h
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : T2Space G
inst✝ : BorelSpace G
H : ContentRegular μ
K : Compacts G
⊢ ∀ (ε : ℝ≥0), 0 < ε → (fun s => ↑(toFun μ s)) K < ⊤ → ↑↑(Content.measure μ) ↑K ≤ (fun s => ↑(toFun μ s)) K + ↑ε

Next tactic:
intro ε εpos _
/-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
  on compact sets. -/
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K := by
  refine' le_antisymm _ _
  · apply ENNReal.le_of_forall_pos_le_add
    intro ε εpos _
    

Proof state:
case refine'_1.h
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : T2Space G
inst✝ : BorelSpace G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : (fun s => ↑(toFun μ s)) K < ⊤
⊢ ↑↑(Content.measure μ) ↑K ≤ (fun s => ↑(toFun μ s)) K + ↑ε

Next tactic:
obtain ⟨K', K'_hyp⟩ := contentRegular_exists_compact μ H K (ne_bot_of_gt εpos)
/-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
  on compact sets. -/
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K := by
  refine' le_antisymm _ _
  · apply ENNReal.le_of_forall_pos_le_add
    intro ε εpos _
    obtain ⟨K', K'_hyp⟩ := contentRegular_exists_compact μ H K (ne_bot_of_gt εpos)
    

Proof state:
case refine'_1.h.intro
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : T2Space G
inst✝ : BorelSpace G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : (fun s => ↑(toFun μ s)) K < ⊤
K' : Compacts G
K'_hyp : K.carrier ⊆ interior K'.carrier ∧ (fun s => ↑(toFun μ s)) K' ≤ (fun s => ↑(toFun μ s)) K + ↑ε
⊢ ↑↑(Content.measure μ) ↑K ≤ (fun s => ↑(toFun μ s)) K + ↑ε

Next tactic:
calc
      μ.measure ↑K ≤ μ.measure (interior ↑K') := by
        rw [μ.measure_apply isOpen_interior.measurableSet,
          μ.measure_apply K.isCompact.measurableSet]
        exact μ.outerMeasure.mono K'_hyp.left
      _ ≤ μ K' := by
        rw [μ.measure_apply (IsOpen.measurableSet isOpen_interior)]
        exact μ.outerMeasure_interior_compacts K'
      _ ≤ μ K + ε := K'_hyp.right
/-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
  on compact sets. -/
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K := by
  refine' le_antisymm _ _
  · apply ENNReal.le_of_forall_pos_le_add
    intro ε εpos _
    obtain ⟨K', K'_hyp⟩ := contentRegular_exists_compact μ H K (ne_bot_of_gt εpos)
    calc
      μ.measure ↑K ≤ μ.measure (interior ↑K') := by
        

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : T2Space G
inst✝ : BorelSpace G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : (fun s => ↑(toFun μ s)) K < ⊤
K' : Compacts G
K'_hyp : K.carrier ⊆ interior K'.carrier ∧ (fun s => ↑(toFun μ s)) K' ≤ (fun s => ↑(toFun μ s)) K + ↑ε
⊢ ↑↑(Content.measure μ) ↑K ≤ ↑↑(Content.measure μ) (interior ↑K')

Next tactic:
rw [μ.measure_apply isOpen_interior.measurableSet,
          μ.measure_apply K.isCompact.measurableSet]
/-- If `μ` is a regular content, then the measure induced by `μ` will agree with `μ`
  on compact sets. -/
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K := by
  refine' le_antisymm _ _
  · apply ENNReal.le_of_forall_pos_le_add
    intro ε εpos _
    obtain ⟨K', K'_hyp⟩ := contentRegular_exists_compact μ H K (ne_bot_of_gt εpos)
    calc
      μ.measure ↑K ≤ μ.measure (interior ↑K') := by
        rw [μ.measure_apply isOpen_interior.measurableSet,
          μ.measure_apply K.isCompact.measurableSet]
        

Proof state:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : T2Space G
inst✝ : BorelSpace G
H : ContentRegular μ
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : (fun s => ↑(toFun μ s)) K < ⊤
K' : Compacts G
K'_hyp : K.carrier ⊆ interior K'.carrier ∧ (fun s => ↑(toFun μ s)) K' ≤ (fun s => ↑(toFun μ s)) K + ↑ε
⊢ ↑(Content.outerMeasure μ) ↑K ≤ ↑(Content.outerMeasure μ) (interior ↑K')

Next tactic:
exact μ.outerMeasure.mono K'_hyp.left