theorem ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable {f : α → E} (hf : Integrable f μ)
    (h'f : StronglyMeasurable f) :
    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by
  /- For every `c`, then `(∫⁻ y in a, ‖f y - c‖₊ ∂μ) / μ a` tends almost everywhere to `‖f x - c‖`.
    We apply this to a countable set of `c` which is dense in the range of `f`, to deduce the
    desired convergence.
    A minor technical inconvenience is that constants are not integrable, so to apply previous
    lemmas we need to replace `c` with the restriction of `c` to a finite measure set `A n` in the
    above sketch. -/
  let A := MeasureTheory.Measure.finiteSpanningSetsInOpen' μ
  rcases h'f.isSeparable_range with ⟨t, t_count, ht⟩
  have main :
    ∀ᵐ x ∂μ,
      ∀ᵉ (n : ℕ) (c ∈ t),
        Tendsto (fun a => (∫⁻ y in a, ‖f y - (A.set n).indicator (fun _ => c) y‖₊ ∂μ) / μ a)
          (v.filterAt x) (𝓝 ‖f x - (A.set n).indicator (fun _ => c) x‖₊) := by
    simp_rw [ae_all_iff, ae_ball_iff t_count]
    intro n c _
    apply ae_tendsto_lintegral_div'
    · refine' (h'f.sub _).ennnorm
      exact stronglyMeasurable_const.indicator (IsOpen.measurableSet (A.set_mem n))
    · apply ne_of_lt
      calc
        (∫⁻ y, ↑‖f y - (A.set n).indicator (fun _ : α => c) y‖₊ ∂μ) ≤
            ∫⁻ y, ‖f y‖₊ + ‖(A.set n).indicator (fun _ : α => c) y‖₊ ∂μ := by
          apply lintegral_mono
          intro x
          dsimp
          rw [← ENNReal.coe_add]
          exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _)
        _ = (∫⁻ y, ‖f y‖₊ ∂μ) + ∫⁻ y, ‖(A.set n).indicator (fun _ : α => c) y‖₊ ∂μ :=
          (lintegral_add_left h'f.ennnorm _)
        _ < ∞ + ∞ :=
          haveI I : Integrable ((A.set n).indicator fun _ : α => c) μ := by
            simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),
              integrableOn_const, A.finite n, or_true_iff]
          ENNReal.add_lt_add hf.2 I.2
  filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x
  have M :
    ∀ c ∈ t, Tendsto (fun a => (∫⁻ y in a, ‖f y - c‖₊ ∂μ) / μ a)
      (v.filterAt x) (𝓝 ‖f x - c‖₊) := by
    intro c hc
    obtain ⟨n, xn⟩ : ∃ n, x ∈ A.set n := by simpa [← A.spanning] using mem_univ x
    specialize hx n c hc
    simp only [xn, indicator_of_mem] at hx
    apply hx.congr' _
    filter_upwards [v.eventually_filterAt_subset_of_nhds (IsOpen.mem_nhds (A.set_mem n) xn),
      v.eventually_filterAt_measurableSet x] with a ha h'a
    congr 1
    apply set_lintegral_congr_fun h'a
    apply eventually_of_forall fun y => ?_
    intro hy
    simp only [ha hy, indicator_of_mem]
  apply ENNReal.tendsto_nhds_zero.2 fun ε εpos => ?_
  obtain ⟨c, ct, xc⟩ : ∃ c ∈ t, (‖f x - c‖₊ : ℝ≥0∞) < ε / 2 := by
    simp_rw [← edist_eq_coe_nnnorm_sub]
    have : f x ∈ closure t := ht (mem_range_self _)
    exact EMetric.mem_closure_iff.1 this (ε / 2) (ENNReal.half_pos (ne_of_gt εpos))
  filter_upwards [(tendsto_order.1 (M c ct)).2 (ε / 2) xc, h'x, v.eventually_measure_lt_top x] with
    a ha h'a h''a
  apply ENNReal.div_le_of_le_mul
  calc
    (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) ≤ ∫⁻ y in a, ‖f y - c‖₊ + ‖f x - c‖₊ ∂μ := by
      apply lintegral_mono fun x => ?_
      

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
h'f : StronglyMeasurable f
A : FiniteSpanningSetsIn μ {K | IsOpen K} := finiteSpanningSetsInOpen' μ
t : Set E
t_count : Set.Countable t
ht : range f ⊆ closure t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto
          (fun a => (∫⁻ (y : α) in a, ↑‖f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y‖₊ ∂μ) / ↑↑μ a)
          (filterAt v x) (𝓝 ↑‖f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x‖₊)
x✝ : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y‖₊ ∂μ) / ↑↑μ a)
        (filterAt v x✝) (𝓝 ↑‖f x✝ - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x✝‖₊)
h'x : ∀ᶠ (a : Set α) in filterAt v x✝, 0 < ↑↑μ a
M : ∀ c ∈ t, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - c‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 ↑‖f x✝ - c‖₊)
ε : ℝ≥0∞
εpos : ε > 0
c : E
ct : c ∈ t
xc : ↑‖f x✝ - c‖₊ < ε / 2
a : Set α
ha : (∫⁻ (y : α) in a, ↑‖f y - c‖₊ ∂μ) / ↑↑μ a < ε / 2
h'a : 0 < ↑↑μ a
h''a : ↑↑μ a < ⊤
x : α
⊢ ↑‖f x - f x✝‖₊ ≤ ↑‖f x - c‖₊ + ↑‖f x✝ - c‖₊

Next tactic:
simpa only [← edist_eq_coe_nnnorm_sub] using edist_triangle_right _ _ _