/-- A sequence of uniformly integrable functions which converges μ-a.e. converges in Lp. -/
theorem tendsto_Lp_of_tendsto_ae_of_meas [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    {f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    (hg' : Memℒp g p μ) (hui : UnifIntegrable f p μ)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0) := by
  rw [ENNReal.tendsto_atTop_zero]
  intro ε hε
  by_cases h : ε < ∞; swap
  · rw [not_lt, top_le_iff] at h
    exact ⟨0, fun n _ => by simp [h]⟩
  by_cases hμ : μ = 0
  · exact ⟨0, fun n _ => by simp [hμ]⟩
  have hε' : 0 < ε.toReal / 3 :=
    div_pos (ENNReal.toReal_pos (gt_iff_lt.1 hε).ne.symm h.ne) (by norm_num)
  have hdivp : 0 ≤ 1 / p.toReal := by
    refine' one_div_nonneg.2 _
    rw [← ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']
    exact le_trans (zero_le _) hp
  have hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal) :=
    Real.rpow_pos_of_pos (measureUnivNNReal_pos hμ) _
  obtain ⟨δ₁, hδ₁, hsnorm₁⟩ := hui hε'
  obtain ⟨δ₂, hδ₂, hsnorm₂⟩ := hg'.snorm_indicator_le μ hp hp' hε'
  obtain ⟨t, htm, ht₁, ht₂⟩ := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg (lt_min hδ₁ hδ₂)
  rw [Metric.tendstoUniformlyOn_iff] at ht₂
  specialize ht₂ (ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)))
    (div_pos (ENNReal.toReal_pos (gt_iff_lt.1 hε).ne.symm h.ne) (mul_pos (by norm_num) hpow))
  obtain ⟨N, hN⟩ := eventually_atTop.1 ht₂; clear ht₂
  refine' ⟨N, fun n hn => _⟩
  rw [← t.indicator_self_add_compl (f n - g)]
  refine' le_trans (snorm_add_le (((hf n).sub hg).indicator htm).aestronglyMeasurable
    (((hf n).sub hg).indicator htm.compl).aestronglyMeasurable hp) _
  rw [sub_eq_add_neg, Set.indicator_add' t, Set.indicator_neg']
  refine' le_trans (add_le_add_right (snorm_add_le ((hf n).indicator htm).aestronglyMeasurable
    (hg.indicator htm).neg.aestronglyMeasurable hp) _) _
  have hnf : snorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3) := by
    refine' hsnorm₁ n t htm (le_trans ht₁ _)
    rw [ENNReal.ofReal_le_ofReal_iff hδ₁.le]
    exact min_le_left _ _
  have hng : snorm (t.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3) := by
    refine' hsnorm₂ t htm (le_trans ht₁ _)
    rw [ENNReal.ofReal_le_ofReal_iff hδ₂.le]
    exact min_le_right _ _
  have hlt : snorm (tᶜ.indicator (f n - g)) p μ ≤ ENNReal.ofReal (ε.toReal / 3) := by
    specialize hN n hn
    have : 0 ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)) := by
      rw [div_mul_eq_div_mul_one_div]
      exact mul_nonneg hε'.le (one_div_nonneg.2 hpow.le)
    have := snorm_sub_le_of_dist_bdd μ hp' htm.compl this fun x hx =>
      (dist_comm (g x) (f n x) ▸ (hN x hx).le :
        dist (f n x) (g x) ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)))
    refine' le_trans this _
    rw [div_mul_eq_div_mul_one_div, ← ENNReal.ofReal_toReal (measure_lt_top μ tᶜ).ne,
      ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hdivp, ← ENNReal.ofReal_mul, mul_assoc]
    · refine' ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right hε'.le _)
      rw [mul_comm, mul_one_div, div_le_one]
      · refine' Real.rpow_le_rpow ENNReal.toReal_nonneg
          (ENNReal.toReal_le_of_le_ofReal (measureUnivNNReal_pos hμ).le _) hdivp
        rw [ENNReal.ofReal_coe_nnreal, coe_measureUnivNNReal]
        

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < measureUnivNNReal μ ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ x ∈ tᶜ, dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ ↑↑μ tᶜ ≤ ↑↑μ univ

Next tactic:
exact measure_mono (Set.subset_univ _)