-- Mathlib/MeasureTheory/Measure/Content.lean
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory

Proof state:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
rw [ENNReal.iSup_add]

Result:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
⊢ ⨆ i, μ ↑i + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Proof state:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'
⊢ μ L' + μ M' ≤ μ.innerContent { carrier := ↑U', is_open' := ⋯ }

Tactic:
refine le_trans (ge_of_eq ?_) (μ.le_innerContent _ _ this)

Result:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'
⊢ μ (L' ⊔ M') = μ L' + μ M'

Proof state:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
⊢ S ≤ μ.outerMeasure.caratheodory

Tactic:
rw [BorelSpace.measurable_eq (α := G)]

Result:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
⊢ borel G ≤ μ.outerMeasure.caratheodory

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
have : ↑U' \ U ⊆ U' \ L' := diff_subset_diff_right hL'U

Result:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Proof state:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩

Result:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
⊢ μ L' + ⨆ x, μ ↑x ≤ μ.outerMeasure ↑U'

Tactic:
rw [ENNReal.add_iSup]

Result:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
⊢ ⨆ i, μ L' + μ ↑i ≤ μ.outerMeasure ↑U'

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
have hL'U' : (L' : Set G) ⊆ (U' : Set G) := IsCompact.closure_subset_of_isOpen L.2 U'.2 hL.1

Result:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Proof state:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
⊢ ∀ (U_1 : Opens G), μ.outerMeasure (↑U_1 ∩ U) + μ.outerMeasure (↑U_1 \ U) ≤ μ.outerMeasure ↑U_1

Tactic:
intro U'

Result:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
⊢ μ.outerMeasure (↑U' ∩ U) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Proof state:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
simp only [subset_inter_iff] at hL

Result:
case mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Proof state:
case mk.mk
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space 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
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'
⊢ μ (L' ⊔ M') = μ L' + μ M'

Tactic:
exact μ.sup_disjoint L' M' (subset_diff.1 hM').2.symm isClosed_closure isClosed_closure

Result:
no goals