open Finset in
/-- In a finite measure space, assume that any open set can be approximated from inside by closed
sets. Then the measure is weakly regular. -/
theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
    (H : InnerRegularWRT μ IsClosed IsOpen) : WeaklyRegular μ := by
  have hfin : ∀ {s}, μ s ≠ ⊤ := @(measure_ne_top μ)
  suffices ∀ s, MeasurableSet s → ∀ ε, ε ≠ 0 → ∃ F, F ⊆ s ∧ ∃ U, U ⊇ s ∧
      IsClosed F ∧ IsOpen U ∧ μ s ≤ μ F + ε ∧ μ U ≤ μ s + ε by
    refine'
      { outerRegular := fun s hs r hr => _
        innerRegular := H }
    rcases exists_between hr with ⟨r', hsr', hr'r⟩
    rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with ⟨-, -, U, hsU, -, hUo, -, H⟩
    refine' ⟨U, hsU, hUo, _⟩
    rw [add_tsub_cancel_of_le hsr'.le] at H
    exact H.trans_lt hr'r
  refine' MeasurableSet.induction_on_open _ _ _
  /- The proof is by measurable induction: we should check that the property is true for the empty
    set, for open sets, and is stable by taking the complement and by taking countable disjoint
    unions. The point of the property we are proving is that it is stable by taking complements
    (exchanging the roles of closed and open sets and thanks to the finiteness of the measure). -/
  -- check for open set
  · intro U hU ε hε
    rcases H.exists_subset_lt_add isClosed_empty hU hfin hε with ⟨F, hsF, hFc, hF⟩
    exact ⟨F, hsF, U, Subset.rfl, hFc, hU, hF.le, le_self_add⟩
  -- check for complements
  · rintro s hs H ε hε
    rcases H ε hε with ⟨F, hFs, U, hsU, hFc, hUo, hF, hU⟩
    refine'
      ⟨Uᶜ, compl_subset_compl.2 hsU, Fᶜ, compl_subset_compl.2 hFs, hUo.isClosed_compl,
        hFc.isOpen_compl, _⟩
    simp only [measure_compl_le_add_iff, *, hUo.measurableSet, hFc.measurableSet, true_and_iff]
  -- check for disjoint unions
  · intro s hsd hsm H ε ε0
    have ε0' : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
    rcases ENNReal.exists_pos_sum_of_countable' ε0' ℕ with ⟨δ, δ0, hδε⟩
    choose F hFs U hsU hFc hUo hF hU using fun n => H n (δ n) (δ0 n).ne'
    -- the approximating closed set is constructed by considering finitely many sets `s i`, which
    -- cover all the measure up to `ε/2`, approximating each of these by a closed set `F i`, and
    -- taking the union of these (finitely many) `F i`.
    have : Tendsto (fun t => (∑ k in t, μ (s k)) + ε / 2) atTop (𝓝 <| μ (⋃ n, s n) + ε / 2) := by
      rw [measure_iUnion hsd hsm]
      exact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds
    rcases (this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin ε0').exists with ⟨t, ht⟩
    -- the approximating open set is constructed by taking for each `s n` an approximating open set
    -- `U n` with measure at most `μ (s n) + δ n` for a summable `δ`, and taking the union of these.
    refine'
      ⟨⋃ k ∈ t, F k, iUnion_mono fun k => iUnion_subset fun _ => hFs _, ⋃ n, U n, iUnion_mono hsU,
        isClosed_biUnion_finset fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _⟩
    · calc
        (∑ k in t, μ (s k)) + ε / 2 ≤ ((∑ k in t, μ (F k)) + ∑ k in t, δ k) + ε / 2 := by
          rw [← sum_add_distrib]
          exact add_le_add_right (sum_le_sum fun k _ => hF k) _
        _ ≤ (∑ k in t, μ (F k)) + ε / 2 + ε / 2 :=
          (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans hδε.le) _) _)
        _ = μ (⋃ k ∈ t, F k) + ε := by
          rw [measure_biUnion_finset, add_assoc, ENNReal.add_halves]
          

Proof state:
case hd
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U✝ s✝ : Set α
ε✝ r : ℝ≥0∞
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : InnerRegularWRT μ IsClosed IsOpen
hfin : ∀ {s : Set α}, ↑↑μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Disjoint on 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 in t, ↑↑μ (s k) + ε / 2) atTop (𝓝 (↑↑μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : ↑↑μ (⋃ n, s n) < ∑ k in t, ↑↑μ (s k) + ε / 2
⊢ PairwiseDisjoint ↑t fun k => F k
case hm
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U✝ s✝ : Set α
ε✝ r : ℝ≥0∞
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : InnerRegularWRT μ IsClosed IsOpen
hfin : ∀ {s : Set α}, ↑↑μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Disjoint on 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 in t, ↑↑μ (s k) + ε / 2) atTop (𝓝 (↑↑μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : ↑↑μ (⋃ n, s n) < ∑ k in t, ↑↑μ (s k) + ε / 2
⊢ ∀ b ∈ t, MeasurableSet (F b)

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