l3lab/ntp-mathlib huggingface dataset:
/-- 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)`. -/
  

Proof state:
case inr.inr.intro.intro.intro
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
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
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 }
/-- 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 }
  

Proof state:
case inr.inr.intro.intro.intro
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) }
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
rcases exist_disjoint_covering_families hτ hN a with ⟨u, hu, hu'⟩
/-- 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'⟩
  

Proof state:
case inr.inr.intro.intro.intro.intro.intro
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)
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
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
/-- 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
    

Proof state:
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)
⊢ ∀ (i : Fin N), Set.Countable (u i)

Next tactic:
intro i
/-- 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
    

Proof state:
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)
i : Fin N
⊢ Set.Countable (u i)

Next tactic:
refine' (hu i).countable_of_nonempty_interior fun j _ => _
/-- 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 _ => _
    

Proof state:
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)
i : Fin N
j : ↑s
x✝ : j ∈ u i
⊢ Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))

Next tactic:
have : (ball (j : α) (r j)).Nonempty := nonempty_ball.2 (a.rpos _)
/-- 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 _)
    

Proof state:
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)
i : Fin N
j : ↑s
x✝ : j ∈ u i
this : Set.Nonempty (ball (↑j) (r ↑j))
⊢ Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))

Next tactic:
exact this.mono ball_subset_interior_closedBall
/-- The **measurable Besicovitch covering theorem**. Assume that, for any `x` in a set `s`,
one is given a set of admissible closed balls centered at `x`, with arbitrarily small radii.
Then there exists a disjoint covering of almost all `s` by admissible closed balls centered at some
points of `s`.
This version requires that the underlying measure is finite, and that the space has the Besicovitch
covering property (which is satisfied for instance by normed real vector spaces). It expresses the
conclusion in a slightly awkward form (with a subset of `α × ℝ`) coming from the proof technique.
For a version assuming that the measure is sigma-finite,
see `exists_disjoint_closedBall_covering_ae_aux`.
For a version giving the conclusion in a nicer form, see `exists_disjoint_closedBall_covering_ae`.
-/
theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measure α)
    [IsFiniteMeasure μ] (f : α → Set ℝ) (s : Set α)
    (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
    ∃ t : Set (α × ℝ),
      t.Countable ∧
        (∀ p : α × ℝ, p ∈ t → p.1 ∈ s) ∧
          (∀ p : α × ℝ, p ∈ t → p.2 ∈ f p.1) ∧
            μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) = 0 ∧
              t.PairwiseDisjoint fun p => closedBall p.1 p.2 := by
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (f x ∩ Ioo 0 δ)
⊢ ∃ t,
    Set.Countable t ∧
      (∀ p ∈ t, p.1 ∈ s) ∧
        (∀ p ∈ t, p.2 ∈ f p.1) ∧
          ↑↑μ (s \ ⋃ p ∈ t, closedBall p.1 p.2) = 0 ∧ PairwiseDisjoint t fun p => closedBall p.1 p.2

Next tactic:
rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩