l3lab/ntp-mathlib huggingface dataset:
theorem tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 (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)
    (u : Set E) (h'u : μ u ≠ 0) (R : ℝ) (Rpos : 0 < R) (t_bound : t ⊆ closedBall 0 R) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 0) := by
  set t' := R⁻¹ • t with ht'
  set u' := R⁻¹ • u with hu'
  have A : Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0) := by
    apply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 μ s x h t' u'
    · simp only [h'u, (pow_pos Rpos _).ne', abs_nonpos_iff, addHaar_smul, not_false_iff,
        ENNReal.ofReal_eq_zero, inv_eq_zero, inv_pow, Ne.def, or_self_iff, mul_eq_zero]
    · refine (smul_set_mono t_bound).trans_eq ?_
      rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le),
        inv_mul_cancel Rpos.ne']
  have B : Tendsto (fun r : ℝ => R * r) (𝓝[>] 0) (𝓝[>] (R * 0)) := by
    apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
    · exact (tendsto_const_nhds.mul tendsto_id).mono_left nhdsWithin_le_nhds
    · filter_upwards [self_mem_nhdsWithin]
      intro r rpos
      rw [mul_zero]
      exact mul_pos Rpos rpos
  rw [mul_zero] at B
  apply (A.comp B).congr' _
  filter_upwards [self_mem_nhdsWithin]
  rintro r -
  have T : (R * r) • t' = r • t := by
    rw [mul_comm, ht', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]
  have U : (R * r) • u' = r • u := by
    rw [mul_comm, hu', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]
  dsimp
  

Proof state:
case h
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 u : Set E
h'u : ↑↑μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
A : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t')) / ↑↑μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
B : Tendsto (fun r => R * r) (𝓝[>] 0) (𝓝[>] 0)
r : ℝ
T : (R * r) • t' = r • t
U : (R * r) • u' = r • u
⊢ ↑↑μ (s ∩ ({x} + (R * r) • R⁻¹ • t)) / ↑↑μ ({x} + (R * r) • R⁻¹ • u) = ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)

Next tactic:
rw [T, U]
/-- 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
  

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 ≠ ⊤
⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)

Next tactic:
refine' tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => _⟩
/-- 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) => _⟩
  

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
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, ↑↑μ (s ∩ ({x} + b • t)) / ↑↑μ ({x} + b • t) < ε

Next tactic:
rcases eq_or_ne (μ t) 0 with (h't | 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)
  · 

Proof state:
case inl
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
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, ↑↑μ (s ∩ ({x} + b • t)) / ↑↑μ ({x} + b • t) < ε

Next tactic:
apply eventually_of_forall fun r => ?_
/-- 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 => ?_
    

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
r : ℝ
⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t) < ε

Next tactic:
suffices H : μ (s ∩ ({x} + r • t)) = 0
/-- 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
    · 

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
r : ℝ
H : ↑↑μ (s ∩ ({x} + r • t)) = 0
⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t) < ε

Next tactic:
rw [H]
/-- 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]; 

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
r : ℝ
H : ↑↑μ (s ∩ ({x} + r • t)) = 0
⊢ 0 / ↑↑μ ({x} + r • t) < ε

Next tactic:
simpa only [ENNReal.zero_div] using εpos
/-- 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
    

Proof state:
case H
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
r : ℝ
⊢ ↑↑μ (s ∩ ({x} + r • t)) = 0

Next tactic:
apply le_antisymm _ (zero_le _)
/-- 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 _)
    

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
r : ℝ
⊢ ↑↑μ (s ∩ ({x} + r • t)) ≤ 0

Next tactic:
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]
/-- 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
        

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
r : ℝ
⊢ ↑↑μ ({x} + r • t) = 0

Next tactic:
simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]
/-- 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]
  

Proof state:
case inr
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
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, ↑↑μ (s ∩ ({x} + b • t)) / ↑↑μ ({x} + b • t) < ε

Next tactic:
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
/-- 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
    

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, 0 < n ∧ ↑↑μ (t \ closedBall 0 ↑n) < ε / 2 * ↑↑μ t

Next tactic:
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))
/-- 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
      

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
⊢ Tendsto (fun n => ↑↑μ (t \ closedBall 0 ↑n)) atTop (𝓝 (↑↑μ (⋂ n, t \ closedBall 0 ↑n)))

Next tactic:
have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne⟩