l3lab/ntp-mathlib huggingface dataset:
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
N : ∃ n, ↑↑μ (t \ closedBall 0 ↑n) ≠ ⊤
⊢ Tendsto (fun n => ↑↑μ (t \ closedBall 0 ↑n)) atTop (𝓝 (↑↑μ (⋂ n, t \ closedBall 0 ↑n)))

Next tactic:
refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
N : ∃ n, ↑↑μ (t \ closedBall 0 ↑n) ≠ ⊤
m n : ℕ
hmn : m ≤ n
⊢ t \ closedBall 0 ↑n ≤ t \ closedBall 0 ↑m

Next tactic:
exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
    

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
A : Tendsto (fun n => ↑↑μ (t \ closedBall 0 ↑n)) atTop (𝓝 (↑↑μ (⋂ n, t \ closedBall 0 ↑n)))
⊢ ∃ n, 0 < n ∧ ↑↑μ (t \ closedBall 0 ↑n) < ε / 2 * ↑↑μ t

Next tactic:
have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
        iUnion_closedBall_nat, compl_univ, inter_empty]
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
    have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
A : Tendsto (fun n => ↑↑μ (t \ closedBall 0 ↑n)) atTop (𝓝 (↑↑μ (⋂ n, t \ closedBall 0 ↑n)))
⊢ ⋂ n, t \ closedBall 0 ↑n = ∅

Next tactic:
simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
        iUnion_closedBall_nat, compl_univ, inter_empty]
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
    have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
        iUnion_closedBall_nat, compl_univ, inter_empty]
    

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
A : Tendsto (fun n => ↑↑μ (t \ closedBall 0 ↑n)) atTop (𝓝 (↑↑μ (⋂ n, t \ closedBall 0 ↑n)))
this : ⋂ n, t \ closedBall 0 ↑n = ∅
⊢ ∃ n, 0 < n ∧ ↑↑μ (t \ closedBall 0 ↑n) < ε / 2 * ↑↑μ t

Next tactic:
simp only [this, measure_empty] at A
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
    have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
        iUnion_closedBall_nat, compl_univ, inter_empty]
    simp only [this, measure_empty] at A
    

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
this : ⋂ n, t \ closedBall 0 ↑n = ∅
A : Tendsto (fun n => ↑↑μ (t \ closedBall 0 ↑n)) atTop (𝓝 0)
⊢ ∃ n, 0 < n ∧ ↑↑μ (t \ closedBall 0 ↑n) < ε / 2 * ↑↑μ t

Next tactic:
have I : 0 < ε / 2 * μ t := ENNReal.mul_pos (ENNReal.half_pos εpos.ne').ne' h't
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
    have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
        iUnion_closedBall_nat, compl_univ, inter_empty]
    simp only [this, measure_empty] at A
    have I : 0 < ε / 2 * μ t := ENNReal.mul_pos (ENNReal.half_pos εpos.ne').ne' h't
    

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
this : ⋂ n, t \ closedBall 0 ↑n = ∅
A : Tendsto (fun n => ↑↑μ (t \ closedBall 0 ↑n)) atTop (𝓝 0)
I : 0 < ε / 2 * ↑↑μ t
⊢ ∃ n, 0 < n ∧ ↑↑μ (t \ closedBall 0 ↑n) < ε / 2 * ↑↑μ t

Next tactic:
exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
    have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
        iUnion_closedBall_nat, compl_univ, inter_empty]
    simp only [this, measure_empty] at A
    have I : 0 < ε / 2 * μ t := ENNReal.mul_pos (ENNReal.half_pos εpos.ne').ne' h't
    exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists
  

Proof state:
case inr.intro.intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
n : ℕ
npos : 0 < n
hn : ↑↑μ (t \ closedBall 0 ↑n) < ε / 2 * ↑↑μ t
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, ↑↑μ (s ∩ ({x} + b • t)) / ↑↑μ ({x} + b • t) < ε

Next tactic:
have L :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • (t ∩ closedBall 0 n))) / μ ({x} + r • t)) (𝓝[>] 0)
      (𝓝 0) :=
    tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 μ s x h _ t h't n (Nat.cast_pos.2 npos)
      (inter_subset_right _ _)
/-- Consider a point `x` at which a set `s` has density zero, with respect to closed balls. Then it
also has density zero with respect to any measurable set `t`: the proportion of points in `s`
belonging to a rescaled copy `{x} + r • t` of `t` tends to zero as `r` tends to zero. -/
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by
  refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
  rcases eq_or_ne (μ t) 0 with (h't | h't)
  · apply eventually_of_forall fun r => ?_
    suffices H : μ (s ∩ ({x} + r • t)) = 0
    · rw [H]; simpa only [ENNReal.zero_div] using εpos
    apply le_antisymm _ (zero_le _)
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono (inter_subset_right _ _)
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
  obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ μ (t \ closedBall 0 n) < ε / 2 * μ t := by
    have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩
      refine' tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ _) N
      exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
    have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
        iUnion_closedBall_nat, compl_univ, inter_empty]
    simp only [this, measure_empty] at A
    have I : 0 < ε / 2 * μ t := ENNReal.mul_pos (ENNReal.half_pos εpos.ne').ne' h't
    exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists
  have L :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • (t ∩ closedBall 0 n))) / μ ({x} + r • t)) (𝓝[>] 0)
      (𝓝 0) :=
    tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 μ s x h _ t h't n (Nat.cast_pos.2 npos)
      (inter_subset_right _ _)
  

Proof state:
case inr.intro.intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : IsAddHaarMeasure μ
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ s : Set E
x : E
h : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : ↑↑μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : ↑↑μ t ≠ 0
n : ℕ
npos : 0 < n
hn : ↑↑μ (t \ closedBall 0 ↑n) < ε / 2 * ↑↑μ t
L : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / ↑↑μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, ↑↑μ (s ∩ ({x} + b • t)) / ↑↑μ ({x} + b • t) < ε

Next tactic:
filter_upwards [(tendsto_order.1 L).2 _ (ENNReal.half_pos εpos.ne'), self_mem_nhdsWithin]