Theorem: MeasureTheory.Measure.OuterRegular.of_restrict
File: Mathlib/MeasureTheory/Measure/Regular.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
⊢ μ.OuterRegular
state_after: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
⊢ ∃ U ⊇ A, IsOpen U ∧ μ U < r
tactic: refine ⟨fun A hA r hr => ?_⟩
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
⊢ ∃ U ⊇ A, IsOpen U ∧ μ U < r
state_after: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ⊤
⊢ ∃ U ⊇ A, IsOpen U ∧ μ U < r
tactic: have HA : μ A < ∞ := lt_of_lt_of_le hr le_top
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ⊤
⊢ ∃ U ⊇ A, IsOpen U ∧ μ U < r
state_after: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ⊤
hm : ∀ (n : ℕ), MeasurableSet (s n)
⊢ ∃ U ⊇ A, IsOpen U ∧ μ U < r
tactic: have hm : ∀ n, MeasurableSet (s n) := fun n => (h' n).measurableSet
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ⊤
hm : ∀ (n : ℕ), MeasurableSet (s n)
⊢ ∃ U ⊇ A, IsOpen U ∧ μ U < r
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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: obtain ⟨A, hAm, hAs, hAd, rfl⟩ :
    ∃ A' : ℕ → Set α,
      (∀ n, MeasurableSet (A' n)) ∧
        (∀ n, A' n ⊆ s n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ n, A' n := by
    refine
      ⟨fun n => A ∩ disjointed s n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
        inter_subset_right.trans (disjointed_subset _ _),
        (disjoint_disjointed s).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, ?_⟩
    rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
tactic: rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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, A n, IsOpen U ∧ μ U < r
tactic: rw [lt_tsub_iff_right, add_comm] at hδε
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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, A n, IsOpen U ∧ μ U < r
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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: have : ∀ n, ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n := by
    intro n
    have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)
    have Ht : μ.restrict (s n) (A n) ≠ ∞ := by
      rw [H₁]
      exact ((measure_mono (inter_subset_left.trans (subset_iUnion A n))).trans_lt HA).ne
    rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
    rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
    exact ⟨U ∩ s n, subset_inter hAU (hAs _), hUo.inter (h' n), hU⟩
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
tactic: choose U hAU hUo hU using this
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure 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
⊢ μ (⋃ n, U n) < r
tactic: refine ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, ?_⟩