-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
theorem sInf_caratheodory (s : Set α) (hs : MeasurableSet s) :
    MeasurableSet[(sInf (toOuterMeasure '' m)).caratheodory] s

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s✝ : Set α
hs : MeasurableSet s✝
t✝ : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t✝ ⊆ u
_hu : MeasurableSet u
s t : Set α
hst : s ⊆ t
⊢ ⨅ b ∈ m, b.toOuterMeasure s ≤ μ t

Tactic:
exact iInf₂_le_of_le μ hμ <| measure_mono hst

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
⊢ MeasurableSet s

Tactic:
refine OuterMeasure.boundedBy_caratheodory fun t => ?_

Result:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) (t ∩ s) + OuterMeasure.sInfGen (toOuterMeasure '' m) (t \ s) ≤
    OuterMeasure.sInfGen (toOuterMeasure '' m) t

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
⊢ ∀ ⦃x : Measure α⦄,
    x ∈ m →
      ∀ (i : Set α),
        t ⊆ i → MeasurableSet i → (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ x i

Tactic:
intro μ hμ u htu _hu

Result:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ u

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
hm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t
⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ u

Tactic:
rw [← measure_inter_add_diff u hs]

Result:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
hm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t
⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ (u ∩ s) + μ (u \ s)

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) (t ∩ s) + OuterMeasure.sInfGen (toOuterMeasure '' m) (t \ s) ≤
    OuterMeasure.sInfGen (toOuterMeasure '' m) t

Tactic:
simp only [OuterMeasure.sInfGen, le_iInf_iff, forall_mem_image, measure_eq_iInf t,
  coe_toOuterMeasure]

Result:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
⊢ ∀ ⦃x : Measure α⦄,
    x ∈ m →
      ∀ (i : Set α),
        t ⊆ i → MeasurableSet i → (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ x i

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
hm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t
⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ (u ∩ s) + μ (u \ s)

Tactic:
exact add_le_add (hm <| inter_subset_inter_left _ htu) (hm <| diff_subset_diff_left htu)

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
⊢ ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t

Tactic:
intro s t hst

Result:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s✝ : Set α
hs : MeasurableSet s✝
t✝ : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t✝ ⊆ u
_hu : MeasurableSet u
s t : Set α
hst : s ⊆ t
⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s✝ : Set α
hs : MeasurableSet s✝
t✝ : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t✝ ⊆ u
_hu : MeasurableSet u
s t : Set α
hst : s ⊆ t
⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t

Tactic:
rw [OuterMeasure.sInfGen_def, iInf_image]

Result:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s✝ : Set α
hs : MeasurableSet s✝
t✝ : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t✝ ⊆ u
_hu : MeasurableSet u
s t : Set α
hst : s ⊆ t
⊢ ⨅ b ∈ m, b.toOuterMeasure s ≤ μ t

Proof state:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ u

Tactic:
have hm : ∀ {s t}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t := by
  intro s t hst
  rw [OuterMeasure.sInfGen_def, iInf_image]
  exact iInf₂_le_of_le μ hμ <| measure_mono hst

Result:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
hm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t
⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ u