-- Mathlib/MeasureTheory/Measure/Regular.lean
theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
    (H : InnerRegularWRT μ IsClosed IsOpen) : WeaklyRegular μ

Proof state:
case bc.h
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed F ∧ IsOpen U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed (F n)
hUo : ∀ (n : ℕ), IsOpen (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t => ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
i✝ : ℕ
a✝ : i✝ ∈ t
⊢ μ (s i✝) ≤ μ (F i✝) + δ i✝

Tactic:
apply hF

Result:
no goals

Proof state:
case hd
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed F ∧ IsOpen U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed (F n)
hUo : ∀ (n : ℕ), IsOpen (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t => ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ (↑t).PairwiseDisjoint F

case hm
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed F ∧ IsOpen U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed (F n)
hUo : ∀ (n : ℕ), IsOpen (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t => ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ ∀ b ∈ t, MeasurableSet (F b)

Tactic:
exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n),
  fun k _ => (hFc k).measurableSet]

Result:
no goals

Proof state:
case iUnion
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
⊢ ∀ (f : ℕ → Set α),
    Pairwise (Function.onFun Disjoint f) →
      (∀ (i : ℕ), MeasurableSet (f i)) →
        (∀ (i : ℕ) (ε : ℝ≥0∞),
            ε ≠ 0 → ∃ F ⊆ f i, ∃ U ⊇ f i, IsClosed F ∧ IsOpen U ∧ μ (f i) ≤ μ F + ε ∧ μ U ≤ μ (f i) + ε) →
          ∀ (ε : ℝ≥0∞),
            ε ≠ 0 →
              ∃ F ⊆ ⋃ i, f i, ∃ U ⊇ ⋃ i, f i, IsClosed F ∧ IsOpen U ∧ μ (⋃ i, f i) ≤ μ F + ε ∧ μ U ≤ μ (⋃ i, f i) + ε

Tactic:
intro s hsd hsm H ε ε0

Result:
case iUnion
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed F ∧ IsOpen U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ F ⊆ ⋃ i, s i, ∃ U ⊇ ⋃ i, s i, IsClosed F ∧ IsOpen U ∧ μ (⋃ i, s i) ≤ μ F + ε ∧ μ U ≤ μ (⋃ i, s i) + ε

Proof state:
case iUnion.intro.intro
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed F ∧ IsOpen U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
⊢ ∃ F ⊆ ⋃ i, s i, ∃ U ⊇ ⋃ i, s i, IsClosed F ∧ IsOpen U ∧ μ (⋃ i, s i) ≤ μ F + ε ∧ μ U ≤ μ (⋃ i, s i) + ε

Tactic:
choose F hFs U hsU hFc hUo hF hU using fun n => H n (δ n) (δ0 n).ne'

Result:
case iUnion.intro.intro
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed F ∧ IsOpen U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed (F n)
hUo : ∀ (n : ℕ), IsOpen (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
⊢ ∃ F ⊆ ⋃ i, s i, ∃ U ⊇ ⋃ i, s i, IsClosed F ∧ IsOpen U ∧ μ (⋃ i, s i) ≤ μ F + ε ∧ μ U ≤ μ (⋃ i, s i) + ε

Proof state:
case iUnion.intro.intro.intro.refine_2
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed F ∧ IsOpen U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed (F n)
hUo : ∀ (n : ℕ), IsOpen (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t => ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ μ (⋃ n, U n) ≤ μ (⋃ i, s i) + ε

Tactic:
calc
  μ (⋃ n, U n) ≤ ∑' n, μ (U n) := measure_iUnion_le _
  _ ≤ ∑' n, (μ (s n) + δ n) := ENNReal.tsum_le_tsum hU
  _ = μ (⋃ n, s n) + ∑' n, δ n := by rw [measure_iUnion hsd hsm, ENNReal.tsum_add]
  _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _

Result:
no goals