l3lab/ntp-mathlib huggingface dataset:
/-- 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
  rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
  /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a
      subset of `s` by admissible balls. -/
  

Proof state:
case intro.intro.intro
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)
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
⊢ ∃ 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:
let P : Finset (α × ℝ) → Prop := fun t =>
    ((t : Set (α × ℝ)).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧
      (∀ p : α × ℝ, p ∈ t → p.1 ∈ s) ∧ ∀ p : α × ℝ, p ∈ t → p.2 ∈ f p.1
/-- 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
  rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
  /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a
      subset of `s` by admissible balls. -/
  let P : Finset (α × ℝ) → Prop := fun t =>
    ((t : Set (α × ℝ)).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧
      (∀ p : α × ℝ, p ∈ t → p.1 ∈ s) ∧ ∀ p : α × ℝ, p ∈ t → p.2 ∈ f p.1
  /- Given a finite good covering of a subset `s`, one can find a larger finite good covering,
    covering additionally a proportion at least `1/(N+1)` of leftover points. This follows from
    `exist_finset_disjoint_balls_large_measure` applied to balls not intersecting the initial
    covering. -/
  have :
      ∀ t : Finset (α × ℝ), P t → ∃ u : Finset (α × ℝ), t ⊆ u ∧ P u ∧
        μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ u), closedBall p.1 p.2) ≤
          N / (N + 1) * μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), 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)
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t => (PairwiseDisjoint ↑t fun p => closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
⊢ ∀ (t : Finset (α × ℝ)),
    P t →
      ∃ u, t ⊆ u ∧ P u ∧ ↑↑μ (s \ ⋃ p ∈ u, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * ↑↑μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)

Next tactic:
intro t ht
/-- 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
  rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
  /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a
      subset of `s` by admissible balls. -/
  let P : Finset (α × ℝ) → Prop := fun t =>
    ((t : Set (α × ℝ)).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧
      (∀ p : α × ℝ, p ∈ t → p.1 ∈ s) ∧ ∀ p : α × ℝ, p ∈ t → p.2 ∈ f p.1
  /- Given a finite good covering of a subset `s`, one can find a larger finite good covering,
    covering additionally a proportion at least `1/(N+1)` of leftover points. This follows from
    `exist_finset_disjoint_balls_large_measure` applied to balls not intersecting the initial
    covering. -/
  have :
      ∀ t : Finset (α × ℝ), P t → ∃ u : Finset (α × ℝ), t ⊆ u ∧ P u ∧
        μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ u), closedBall p.1 p.2) ≤
          N / (N + 1) * μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) := by
    intro t ht
    

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)
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t => (PairwiseDisjoint ↑t fun p => closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
⊢ ∃ u, t ⊆ u ∧ P u ∧ ↑↑μ (s \ ⋃ p ∈ u, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * ↑↑μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)

Next tactic:
set B := ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2 with hB
/-- 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
  rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
  /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a
      subset of `s` by admissible balls. -/
  let P : Finset (α × ℝ) → Prop := fun t =>
    ((t : Set (α × ℝ)).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧
      (∀ p : α × ℝ, p ∈ t → p.1 ∈ s) ∧ ∀ p : α × ℝ, p ∈ t → p.2 ∈ f p.1
  /- Given a finite good covering of a subset `s`, one can find a larger finite good covering,
    covering additionally a proportion at least `1/(N+1)` of leftover points. This follows from
    `exist_finset_disjoint_balls_large_measure` applied to balls not intersecting the initial
    covering. -/
  have :
      ∀ t : Finset (α × ℝ), P t → ∃ u : Finset (α × ℝ), t ⊆ u ∧ P u ∧
        μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ u), closedBall p.1 p.2) ≤
          N / (N + 1) * μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) := by
    intro t ht
    set B := ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2 with hB
    

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)
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t => (PairwiseDisjoint ↑t fun p => closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
⊢ ∃ u, t ⊆ u ∧ P u ∧ ↑↑μ (s \ ⋃ p ∈ u, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * ↑↑μ (s \ B)

Next tactic:
have B_closed : IsClosed B := isClosed_biUnion_finset fun i _ => isClosed_ball
/-- 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
  rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
  /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a
      subset of `s` by admissible balls. -/
  let P : Finset (α × ℝ) → Prop := fun t =>
    ((t : Set (α × ℝ)).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧
      (∀ p : α × ℝ, p ∈ t → p.1 ∈ s) ∧ ∀ p : α × ℝ, p ∈ t → p.2 ∈ f p.1
  /- Given a finite good covering of a subset `s`, one can find a larger finite good covering,
    covering additionally a proportion at least `1/(N+1)` of leftover points. This follows from
    `exist_finset_disjoint_balls_large_measure` applied to balls not intersecting the initial
    covering. -/
  have :
      ∀ t : Finset (α × ℝ), P t → ∃ u : Finset (α × ℝ), t ⊆ u ∧ P u ∧
        μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ u), closedBall p.1 p.2) ≤
          N / (N + 1) * μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) := by
    intro t ht
    set B := ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2 with hB
    have B_closed : IsClosed B := isClosed_biUnion_finset fun i _ => isClosed_ball
    

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)
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t => (PairwiseDisjoint ↑t fun p => closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed B
⊢ ∃ u, t ⊆ u ∧ P u ∧ ↑↑μ (s \ ⋃ p ∈ u, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * ↑↑μ (s \ B)

Next tactic:
set s' := s \ B
/-- 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
  rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
  /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a
      subset of `s` by admissible balls. -/
  let P : Finset (α × ℝ) → Prop := fun t =>
    ((t : Set (α × ℝ)).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧
      (∀ p : α × ℝ, p ∈ t → p.1 ∈ s) ∧ ∀ p : α × ℝ, p ∈ t → p.2 ∈ f p.1
  /- Given a finite good covering of a subset `s`, one can find a larger finite good covering,
    covering additionally a proportion at least `1/(N+1)` of leftover points. This follows from
    `exist_finset_disjoint_balls_large_measure` applied to balls not intersecting the initial
    covering. -/
  have :
      ∀ t : Finset (α × ℝ), P t → ∃ u : Finset (α × ℝ), t ⊆ u ∧ P u ∧
        μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ u), closedBall p.1 p.2) ≤
          N / (N + 1) * μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) := by
    intro t ht
    set B := ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2 with hB
    have B_closed : IsClosed B := isClosed_biUnion_finset fun i _ => isClosed_ball
    set s' := s \ B
    have : ∀ x ∈ s', ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r) := 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)
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t => (PairwiseDisjoint ↑t fun p => closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed B
s' : Set α := s \ B
⊢ ∀ x ∈ s', ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)

Next tactic:
intro x hx
/-- 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 sigma-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 giving the conclusion in a nicer form, see `exists_disjoint_closedBall_covering_ae`.
-/
theorem exists_disjoint_closedBall_covering_ae_aux (μ : Measure α) [SigmaFinite 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
  /- This is deduced from the finite measure case, by using a finite measure with respect to which
    the initial sigma-finite measure is absolutely continuous. -/
  

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✝ : SigmaFinite 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 exists_absolutelyContinuous_isFiniteMeasure μ with ⟨ν, hν, hμν⟩
/-- 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 sigma-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 giving the conclusion in a nicer form, see `exists_disjoint_closedBall_covering_ae`.
-/
theorem exists_disjoint_closedBall_covering_ae_aux (μ : Measure α) [SigmaFinite 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
  /- This is deduced from the finite measure case, by using a finite measure with respect to which
    the initial sigma-finite measure is absolutely continuous. -/
  rcases exists_absolutelyContinuous_isFiniteMeasure μ with ⟨ν, hν, hμν⟩
  

Proof state:
case intro.intro
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✝ : SigmaFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (f x ∩ Ioo 0 δ)
ν : Measure α
hν : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ ∃ 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 exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux ν f s hf with
    ⟨t, t_count, ts, tr, tν, tdisj⟩
/-- 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 sigma-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 giving the conclusion in a nicer form, see `exists_disjoint_closedBall_covering_ae`.
-/
theorem exists_disjoint_closedBall_covering_ae_aux (μ : Measure α) [SigmaFinite 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
  /- This is deduced from the finite measure case, by using a finite measure with respect to which
    the initial sigma-finite measure is absolutely continuous. -/
  rcases exists_absolutelyContinuous_isFiniteMeasure μ with ⟨ν, hν, hμν⟩
  rcases exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux ν f s hf with
    ⟨t, t_count, ts, tr, tν, tdisj⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
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✝ : SigmaFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (f x ∩ Ioo 0 δ)
ν : Measure α
hν : IsFiniteMeasure ν
hμν : μ ≪ ν
t : Set (α × ℝ)
t_count : Set.Countable t
ts : ∀ p ∈ t, p.1 ∈ s
tr : ∀ p ∈ t, p.2 ∈ f p.1
tν : ↑↑ν (s \ ⋃ p ∈ t, closedBall p.1 p.2) = 0
tdisj : PairwiseDisjoint t fun p => closedBall p.1 p.2
⊢ ∃ 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:
exact ⟨t, t_count, ts, tr, hμν tν, tdisj⟩