-- Mathlib/MeasureTheory/Measure/Regular.lean
/-- If the restrictions of a measure to countably many open sets covering the space are
outer regular, then the measure itself is outer regular. -/
lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set α}
    (h : ∀ n, OuterRegular (μ.restrict (s n))) (h' : ∀ n, IsOpen (s n)) (h'' : univ ⊆ ⋃ n, s n) :
    OuterRegular μ

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
⊢ ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r

Tactic:
rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩

Result:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < r - μ (⋃ n, A n)
⊢ ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r

Proof state:
case intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
Ht : (μ.restrict (s n)) (A n) ≠ ⊤
U : Set α
hAU : U ⊇ A n
hUo : IsOpen U
hU : (μ.restrict (s n)) U < (μ.restrict (s n)) (A n) + δ n
⊢ ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n

Tactic:
rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU

Result:
case intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
Ht : (μ.restrict (s n)) (A n) ≠ ⊤
U : Set α
hAU : U ⊇ A n
hUo : IsOpen U
hU : μ (U ∩ s n) < μ (A n) + δ n
⊢ ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n

Proof state:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
⊢ ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n

Tactic:
have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)

Result:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
⊢ ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n

Proof state:
case intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
Ht : (μ.restrict (s n)) (A n) ≠ ⊤
U : Set α
hAU : U ⊇ A n
hUo : IsOpen U
hU : μ (U ∩ s n) < μ (A n) + δ n
⊢ ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n

Tactic:
exact ⟨U ∩ s n, subset_inter hAU (hAs _), hUo.inter (h' n), hU⟩

Result:
no goals

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
this : ∀ (n : ℕ), ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n
⊢ ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r

Tactic:
choose U hAU hUo hU using this

Result:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
U : ℕ → Set α
hAU : ∀ (n : ℕ), U n ⊇ A n
hUo : ∀ (n : ℕ), IsOpen (U n)
hU : ∀ (n : ℕ), μ (U n) < μ (A n) + δ n
⊢ ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r