theorem tendsto_lintegral_norm_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ)
    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0) := by
  have f_measurable : AEStronglyMeasurable f μ :=
    aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim
  let b a := 2 * ENNReal.ofReal (bound a)
  /- `‖F n a‖ ≤ bound a` and `F n a --> f a` implies `‖f a‖ ≤ bound a`, and thus by the
    triangle inequality, have `‖F n a - f a‖ ≤ 2 * (bound a)`. -/
  have hb : ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a := by
    intro n
    filter_upwards [all_ae_ofReal_F_le_bound h_bound n,
      all_ae_ofReal_f_le_bound h_bound h_lim] with a h₁ h₂
    calc
      ENNReal.ofReal ‖F n a - f a‖ ≤ ENNReal.ofReal ‖F n a‖ + ENNReal.ofReal ‖f a‖ := by
        rw [← ENNReal.ofReal_add]
        apply ofReal_le_ofReal
        · apply norm_sub_le
        · exact norm_nonneg _
        · exact norm_nonneg _
      _ ≤ ENNReal.ofReal (bound a) + ENNReal.ofReal (bound a) := (add_le_add h₁ h₂)
      _ = b a := by rw [← two_mul]
  -- On the other hand, `F n a --> f a` implies that `‖F n a - f a‖ --> 0`
  have h : ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0) := by
    rw [← ENNReal.ofReal_zero]
    refine' h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp _
    rwa [← tendsto_iff_norm_sub_tendsto_zero]
  /- Therefore, by the dominated convergence theorem for nonnegative integration, have
    ` ∫ ‖f a - F n a‖ --> 0 ` -/
  suffices h : Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 (∫⁻ _ : α, 0 ∂μ))
  · rwa [lintegral_zero] at h
  -- Using the dominated convergence theorem.
  refine' tendsto_lintegral_of_dominated_convergence' _ _ hb _ _
  -- Show `fun a => ‖f a - F n a‖` is almost everywhere measurable for all `n`
  · exact fun n =>
      measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable
  -- Show `2 * bound` `HasFiniteIntegral`
  · rw [hasFiniteIntegral_iff_ofReal] at bound_hasFiniteIntegral
    · 

Proof state:
case h.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
F✝ : ℕ → α → β
f✝ : α → β
bound✝ : α → ℝ
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : ∫⁻ (a : α), ENNReal.ofReal (bound a) ∂μ < ⊤
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a => 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
⊢ ∫⁻ (a : α), b a ∂μ ≠ ⊤

Next tactic:
calc
        ∫⁻ a, b a ∂μ = 2 * ∫⁻ a, ENNReal.ofReal (bound a) ∂μ := by
          rw [lintegral_const_mul']
          exact coe_ne_top
        _ ≠ ∞ := mul_ne_top coe_ne_top bound_hasFiniteIntegral.ne