/-- 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 _ _)
  filter_upwards [(tendsto_order.1 L).2 _ (ENNReal.half_pos εpos.ne'), self_mem_nhdsWithin]
  rintro r hr (rpos : 0 < r)
  have I :
    μ (s ∩ ({x} + r • t)) ≤
      μ (s ∩ ({x} + r • (t ∩ closedBall 0 n))) + μ ({x} + r • (t \ closedBall 0 n)) :=
    calc
      μ (s ∩ ({x} + r • t)) =
          μ (s ∩ ({x} + r • (t ∩ closedBall 0 n)) ∪ s ∩ ({x} + r • (t \ closedBall 0 n))) :=
        by rw [← inter_union_distrib_left, ← add_union, ← smul_set_union, inter_union_diff]
      _ ≤ μ (s ∩ ({x} + r • (t ∩ closedBall 0 n))) + μ (s ∩ ({x} + r • (t \ closedBall 0 n))) :=
        (measure_union_le _ _)
      _ ≤ μ (s ∩ ({x} + r • (t ∩ closedBall 0 n))) + μ ({x} + r • (t \ closedBall 0 n)) :=
        add_le_add le_rfl (measure_mono (inter_subset_right _ _))
  

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
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)
r : ℝ
hr : ↑↑μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / ↑↑μ ({x} + r • t) < ε / 2
rpos : 0 < r
I : ↑↑μ (s ∩ ({x} + r • t)) ≤ ↑↑μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) + ↑↑μ ({x} + r • (t \ closedBall 0 ↑n))
⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t) < ε

Next tactic:
calc
    μ (s ∩ ({x} + r • t)) / μ ({x} + r • t) ≤
        (μ (s ∩ ({x} + r • (t ∩ closedBall 0 n))) + μ ({x} + r • (t \ closedBall 0 n))) /
          μ ({x} + r • t) :=
      mul_le_mul_right' I _
    _ < ε / 2 + ε / 2 := by
      rw [ENNReal.add_div]
      apply ENNReal.add_lt_add hr _
      rwa [addHaar_singleton_add_smul_div_singleton_add_smul μ rpos.ne',
        ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)]
    _ = ε := ENNReal.add_halves _