/-- The measurable Vitali covering theorem. Assume one is given a family `t` of closed sets with
nonempty interior, such that each `a ∈ t` is included in a ball `B (x, r)` and covers a definite
proportion of the ball `B (x, 3 r)` for a given measure `μ` (think of the situation where `μ` is
a doubling measure and `t` is a family of balls). Consider a (possibly non-measurable) set `s`
at which the family is fine, i.e., every point of `s` belongs to arbitrarily small elements of `t`.
Then one can extract from `t` a disjoint subfamily that covers almost all `s`.

For more flexibility, we give a statement with a parameterized family of sets.
-/
theorem exists_disjoint_covering_ae [MetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
    [SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
    (C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
    (μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ C * μ (B a))
    (ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a))
    (hf : ∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ t, r a ≤ ε ∧ c a = x) :
    ∃ (u : _) (_ : u ⊆ t), u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0 := by
  /- The idea of the proof is the following. Assume for simplicity that `μ` is finite. Applying the
  abstract Vitali covering theorem with `δ = r` given by `hf`, one obtains a disjoint subfamily `u`,
  such that any element of `t` intersects an element of `u` with comparable radius. Fix `ε > 0`.
  Since the elements of `u` have summable measure, one can remove finitely elements `w_1, ..., w_n`.
  so that the measure of the remaining elements is `< ε`. Consider now a point `z` not
  in the `w_i`. There is a small ball around `z` not intersecting the `w_i` (as they are closed),
  an element `a ∈ t` contained in this small ball (as the family `t` is fine at `z`) and an element
  `b ∈ u` intersecting `a`, with comparable radius (by definition of `u`). Then `z` belongs to the
  enlargement of `b`. This shows that `s \ (w_1 ∪ ... ∪ w_n)` is contained in
  `⋃ (b ∈ u \ {w_1, ... w_n}) (enlargement of b)`. The measure of the latter set is bounded by
  `∑ (b ∈ u \ {w_1, ... w_n}) C * μ b` (by the doubling property of the measure), which is at most
  `C ε`. Letting `ε` tend to `0` shows that `s` is almost everywhere covered by the family `u`.

  For the real argument, the measure is only locally finite. Therefore, we implement the same
  strategy, but locally restricted to balls on which the measure is finite. For this, we do not
  use the whole family `t`, but a subfamily `t'` supported on small balls (which is possible since
  the family is assumed to be fine at every point of `s`).
  -/
  -- choose around each `x` a small ball on which the measure is finite
  have : ∀ x, ∃ R, 0 < R ∧ R ≤ 1 ∧ μ (closedBall x (20 * R)) < ∞ := by
    intro x
    obtain ⟨R, Rpos, μR⟩ : ∃ R, 0 < R ∧ μ (closedBall x R) < ∞ :=
      (μ.finiteAt_nhds x).exists_mem_basis nhds_basis_closedBall
    refine' ⟨min 1 (R / 20), _, min_le_left _ _, _⟩
    · simp only [true_and_iff, lt_min_iff, zero_lt_one]
      linarith
    · 

Proof state:
case intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ a ∈ t, Set.Nonempty (interior (B a))
h't : ∀ a ∈ t, IsClosed (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ ↑↑μ (closedBall x (20 * min 1 (R / 20))) < ⊤

Next tactic:
apply lt_of_le_of_lt (measure_mono _) μR