Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
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)

Tactic:
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)
  · filter_upwards with r
    suffices H : μ (s ∩ ({x} + r • t)) = 0 by
      rw [H]; simpa only [ENNReal.zero_div] using εpos
    rw [← nonpos_iff_eq_zero]
    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 sdiff_subset).trans_lt h''t.lt_top).ne⟩
      refine tendsto_measure_iInter_atTop
        (fun n ↦ (ht.diff measurableSet_closedBall).nullMeasurableSet) (fun m n hmn ↦ ?_) N
      exact sdiff_subset_sdiff Subset.rfl (by gcongr)
    have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
      simp_rw [sdiff_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_sdiff]
      _ ≤ 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)) := by
        gcongr; apply inter_subset_right
  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) := by gcongr
    _ < 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 _

Goal state after: