l3lab/ntp-mathlib huggingface dataset:
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL : ↑L ⊆ ↑U' ∩ U
⊢ ↑(toFun μ ↑{ val := L, property := hL }) + ↑(Content.outerMeasure μ) (↑U' \ U) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
simp only [subset_inter_iff] at hL
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
⊢ ↑(toFun μ ↑{ val := L, property := hL✝ }) + ↑(Content.outerMeasure μ) (↑U' \ U) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this : ↑U' \ U ⊆ ↑U' \ ↑L
⊢ ↑(toFun μ ↑{ val := L, property := hL✝ }) + ↑(Content.outerMeasure μ) (↑U' \ U) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this : ↑U' \ U ⊆ ↑U' \ ↑L
⊢ ↑(toFun μ ↑{ val := L, property := hL✝ }) + ↑(Content.outerMeasure μ) (↑U' \ ↑L) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this : ↑U' \ U ⊆ ↑U' \ ↑L
⊢ ↑(toFun μ ↑{ val := L, property := hL✝ }) +
      innerContent μ { carrier := ↑U' \ ↑L, is_open' := (_ : IsOpen (↑U' \ ↑L)) } ≤
    ↑(Content.outerMeasure μ) ↑U'

Next tactic:
simp only [innerContent, iSup_subtype']
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this : ↑U' \ U ⊆ ↑U' \ ↑L
⊢ ↑(toFun μ L) + ⨆ x, ↑(toFun μ ↑x) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
rw [Opens.coe_mk]
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this : ↑U' \ U ⊆ ↑U' \ ↑L
⊢ ↑(toFun μ L) + ⨆ x, ↑(toFun μ ↑x) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L
this : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
⊢ ↑(toFun μ L) + ⨆ x, ↑(toFun μ ↑x) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
rw [ENNReal.add_iSup]
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L
this : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
⊢ ⨆ b, ↑(toFun μ L) + ↑(toFun μ ↑b) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
refine' iSup_le _
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  refine' iSup_le _
  

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L
this : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
⊢ ∀ (i : { i // ↑i ⊆ ↑U' \ ↑L }), ↑(toFun μ L) + ↑(toFun μ ↑i) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
rintro ⟨M, hM⟩
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  refine' iSup_le _
  rintro ⟨M, hM⟩
  

Proof state:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L
this : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L
⊢ ↑(toFun μ L) + ↑(toFun μ ↑{ val := M, property := hM }) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
simp only [subset_diff] at hM
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  refine' iSup_le _
  rintro ⟨M, hM⟩
  simp only [subset_diff] at hM
  

Proof state:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L
this : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
M : Compacts G
hM✝ : ↑M ⊆ ↑U' \ ↑L
hM : ↑M ⊆ ↑U' ∧ Disjoint ↑M ↑L
⊢ ↑(toFun μ L) + ↑(toFun μ ↑{ val := M, property := hM✝ }) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
have : (↑(L ⊔ M) : Set G) ⊆ U' := by
    simp only [union_subset_iff, Compacts.coe_sup, hM, hL, and_self_iff]
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  refine' iSup_le _
  rintro ⟨M, hM⟩
  simp only [subset_diff] at hM
  have : (↑(L ⊔ M) : Set G) ⊆ U' := by
    

Proof state:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L
this : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
M : Compacts G
hM✝ : ↑M ⊆ ↑U' \ ↑L
hM : ↑M ⊆ ↑U' ∧ Disjoint ↑M ↑L
⊢ ↑(L ⊔ M) ⊆ ↑U'

Next tactic:
simp only [union_subset_iff, Compacts.coe_sup, hM, hL, and_self_iff]
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  refine' iSup_le _
  rintro ⟨M, hM⟩
  simp only [subset_diff] at hM
  have : (↑(L ⊔ M) : Set G) ⊆ U' := by
    simp only [union_subset_iff, Compacts.coe_sup, hM, hL, and_self_iff]
  

Proof state:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
M : Compacts G
hM✝ : ↑M ⊆ ↑U' \ ↑L
hM : ↑M ⊆ ↑U' ∧ Disjoint ↑M ↑L
this : ↑(L ⊔ M) ⊆ ↑U'
⊢ ↑(toFun μ L) + ↑(toFun μ ↑{ val := M, property := hM✝ }) ≤ ↑(Content.outerMeasure μ) ↑U'

Next tactic:
rw [μ.outerMeasure_of_isOpen (↑U') U'.2]
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  refine' iSup_le _
  rintro ⟨M, hM⟩
  simp only [subset_diff] at hM
  have : (↑(L ⊔ M) : Set G) ⊆ U' := by
    simp only [union_subset_iff, Compacts.coe_sup, hM, hL, and_self_iff]
  rw [μ.outerMeasure_of_isOpen (↑U') U'.2]
  

Proof state:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
M : Compacts G
hM✝ : ↑M ⊆ ↑U' \ ↑L
hM : ↑M ⊆ ↑U' ∧ Disjoint ↑M ↑L
this : ↑(L ⊔ M) ⊆ ↑U'
⊢ ↑(toFun μ L) + ↑(toFun μ ↑{ val := M, property := hM✝ }) ≤
    innerContent μ { carrier := ↑U', is_open' := (_ : IsOpen U'.carrier) }

Next tactic:
refine' le_trans (ge_of_eq _) (μ.le_innerContent _ _ this)
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
  rw [@BorelSpace.measurable_eq G _ _]
  refine' MeasurableSpace.generateFrom_le _
  intro U hU
  rw [μ.outerMeasure_caratheodory]
  intro U'
  rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.iSup_add]
  refine' iSup_le _
  rintro ⟨L, hL⟩
  simp only [subset_inter_iff] at hL
  have : ↑U' \ U ⊆ U' \ L := diff_subset_diff_right hL.2
  refine' le_trans (add_le_add_left (μ.outerMeasure.mono' this) _) _
  rw [μ.outerMeasure_of_isOpen (↑U' \ L) (IsOpen.sdiff U'.2 L.2.isClosed)]
  simp only [innerContent, iSup_subtype']
  rw [Opens.coe_mk]
  haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ L } := ⟨⟨⊥, empty_subset _⟩⟩
  rw [ENNReal.add_iSup]
  refine' iSup_le _
  rintro ⟨M, hM⟩
  simp only [subset_diff] at hM
  have : (↑(L ⊔ M) : Set G) ⊆ U' := by
    simp only [union_subset_iff, Compacts.coe_sup, hM, hL, and_self_iff]
  rw [μ.outerMeasure_of_isOpen (↑U') U'.2]
  refine' le_trans (ge_of_eq _) (μ.le_innerContent _ _ this)
  

Proof state:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : T2Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL✝ : ↑L ⊆ ↑U' ∩ U
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ ↑L }
M : Compacts G
hM✝ : ↑M ⊆ ↑U' \ ↑L
hM : ↑M ⊆ ↑U' ∧ Disjoint ↑M ↑L
this : ↑(L ⊔ M) ⊆ ↑U'
⊢ (fun s => ↑(toFun μ s)) (L ⊔ M) = ↑(toFun μ L) + ↑(toFun μ ↑{ val := M, property := hM✝ })

Next tactic:
exact μ.sup_disjoint _ _ hM.2.symm
/-- In a locally compact space, any measure constructed from a content is regular. -/
instance regular [WeaklyLocallyCompactSpace G] : μ.measure.Regular := 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
⊢ Measure.Regular (Content.measure μ)

Next tactic:
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⟩
/-- 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
    

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
⊢ Measure.OuterRegular (Content.measure μ)

Next tactic:
refine' ⟨fun A hA r (hr : _ < _) => _⟩