/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
    exact (lt_irrefl _ hμs).elim
  have Npos : N ≠ 0 := by
    rintro rfl
    inhabit α
    exact not_isEmpty_of_nonempty _ hN
  -- introduce a measurable superset `o` with the same measure, for measure computations
  obtain ⟨o, so, omeas, μo⟩ : ∃ o : Set α, s ⊆ o ∧ MeasurableSet o ∧ μ o = μ s :=
    exists_measurable_superset μ s
  /- We will apply the topological Besicovitch theorem, giving `N` disjoint subfamilies of balls
    covering `s`. Among these, one of them covers a proportion at least `1/N` of `s`. A large
    enough finite subfamily will then cover a proportion at least `1/(N+1)`. -/
  let a : BallPackage s α :=
    { c := fun x => x
      r := fun x => r x
      rpos := fun x => rpos x x.2
      r_bound := 1
      r_le := fun x => rle x x.2 }
  rcases exist_disjoint_covering_families hτ hN a with ⟨u, hu, hu'⟩
  have u_count : ∀ i, (u i).Countable := by
    intro i
    refine' (hu i).countable_of_nonempty_interior fun j _ => _
    have : (ball (j : α) (r j)).Nonempty := nonempty_ball.2 (a.rpos _)
    exact this.mono ball_subset_interior_closedBall
  let v : Fin N → Set α := fun i => ⋃ (x : s) (_ : x ∈ u i), closedBall x (r x)
  have A : s = ⋃ i : Fin N, s ∩ v i := by
    refine' Subset.antisymm _ (iUnion_subset fun i => inter_subset_left _ _)
    intro x hx
    obtain ⟨i, y, hxy, h'⟩ :
        ∃ (i : Fin N) (i_1 : ↥s), i_1 ∈ u i ∧ x ∈ ball (↑i_1) (r ↑i_1) := by
      have : x ∈ range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]
      simpa only [mem_iUnion, bex_def] using hu' this
    refine' mem_iUnion.2 ⟨i, ⟨hx, _⟩⟩
    simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk]
    exact ⟨y, ⟨y.2, by simpa only [Subtype.coe_eta]⟩, ball_subset_closedBall h'⟩
  have S : ∑ _i : Fin N, μ s / N ≤ ∑ i, μ (s ∩ v i) :=
    calc
      ∑ _i : Fin N, μ s / N = μ s := by
        simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul]
        rw [ENNReal.mul_div_cancel']
        · simp only [Npos, Ne.def, Nat.cast_eq_zero, not_false_iff]
        · exact ENNReal.nat_ne_top _
      _ ≤ ∑ i, μ (s ∩ v i) := by
        conv_lhs => rw [A]
        apply measure_iUnion_fintype_le
  -- choose an index `i` of a subfamily covering at least a proportion `1/N` of `s`.
  obtain ⟨i, -, hi⟩ : ∃ (i : Fin N), i ∈ Finset.univ ∧ μ s / N ≤ μ (s ∩ v i) := by
    apply ENNReal.exists_le_of_sum_le _ S
    exact ⟨⟨0, bot_lt_iff_ne_bot.2 Npos⟩, Finset.mem_univ _⟩
  replace hi : μ s / (N + 1) < μ (s ∩ v i)
  · apply lt_of_lt_of_le _ hi
    apply (ENNReal.mul_lt_mul_left hμs.ne' (measure_lt_top μ s).ne).2
    rw [ENNReal.inv_lt_inv]
    conv_lhs => rw [← add_zero (N : ℝ≥0∞)]
    exact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one
  have B : μ (o ∩ v i) = ∑' x : u i, μ (o ∩ closedBall x (r x)) := by
    have : o ∩ v i = ⋃ (x : s) (_ : x ∈ u i), o ∩ closedBall x (r x) := by simp only [inter_iUnion]
    rw [this, measure_biUnion (u_count i)]
    · exact (hu i).mono fun k => inter_subset_right _ _
    · exact fun b _ => omeas.inter measurableSet_closedBall
  -- A large enough finite subfamily of `u i` will also cover a proportion `> 1/(N+1)` of `s`.
  -- Since `s` might not be measurable, we express this in terms of the measurable superset `o`.
  obtain ⟨w, hw⟩ :
    ∃ w : Finset (u i), μ s / (N + 1) <
      ∑ x : u i in w, μ (o ∩ closedBall (x : α) (r (x : α))) := by
    have C : HasSum (fun x : u i => μ (o ∩ closedBall x (r x))) (μ (o ∩ v i)) := by
      rw [B]; exact ENNReal.summable.hasSum
    have : μ s / (N + 1) < μ (o ∩ v i) := hi.trans_le (measure_mono (inter_subset_inter_left _ so))
    exact ((tendsto_order.1 C).1 _ this).exists
  -- Bring back the finset `w i` of `↑(u i)` to a finset of `α`, and check that it works by design.
  refine' ⟨Finset.image (fun x : u i => x) w, _, _, _⟩
  -- show that the finset is included in `s`.
  · simp only [image_subset_iff, Finset.coe_image]
    intro y _
    simp only [Subtype.coe_prop, mem_preimage]
  -- show that it covers a large enough proportion of `s`. For measure computations, we do not
  -- use `s` (which might not be measurable), but its measurable superset `o`. Since their measures
  -- are the same, this does not spoil the estimates
  · suffices H : μ (o \ ⋃ x ∈ w, closedBall (↑x) (r ↑x)) ≤ N / (N + 1) * μ s
    · rw [Finset.set_biUnion_finset_image]
      

Proof state:
case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
h✝ : Nonempty α
Npos : N ≠ 0
o : Set α
so : s ⊆ o
omeas : MeasurableSet o
μo : ↑↑μ o = ↑↑μ s
a : BallPackage (↑s) α :=
  { c := fun x => ↑x, r := fun x => r ↑x, rpos := (_ : ∀ (x : ↑s), 0 < r ↑x), r_bound := 1,
    r_le := (_ : ∀ (x : ↑s), r ↑x ≤ 1) }
u : Fin N → Set ↑s
hu : ∀ (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)
hu' : range a.c ⊆ ⋃ i, ⋃ j ∈ u i, ball (BallPackage.c a j) (BallPackage.r a j)
u_count : ∀ (i : Fin N), Set.Countable (u i)
v : Fin N → Set α := fun i => ⋃ x ∈ u i, closedBall (↑x) (r ↑x)
A : s = ⋃ i, s ∩ v i
S : ∑ _i : Fin N, ↑↑μ s / ↑N ≤ ∑ i : Fin N, ↑↑μ (s ∩ v i)
i : Fin N
hi : ↑↑μ s / (↑N + 1) < ↑↑μ (s ∩ v i)
B : ↑↑μ (o ∩ v i) = ∑' (x : ↑(u i)), ↑↑μ (o ∩ closedBall (↑↑x) (r ↑↑x))
w : Finset ↑(u i)
hw : ↑↑μ s / (↑N + 1) < ∑ x in w, ↑↑μ (o ∩ closedBall (↑↑x) (r ↑↑x))
H : ↑↑μ (o \ ⋃ x ∈ w, closedBall (↑↑x) (r ↑↑x)) ≤ ↑N / (↑N + 1) * ↑↑μ s
⊢ ↑↑μ (s \ ⋃ y ∈ w, closedBall (↑↑y) (r ↑↑y)) ≤ ↑N / (↑N + 1) * ↑↑μ s

Next tactic:
exact le_trans (measure_mono (diff_subset_diff so (Subset.refl _))) H