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]
        

Proof state:
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 : HasFiniteIntegral bound
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)
n : ℕ
a : α
h₁ : ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
h₂ : ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)
⊢ ENNReal.ofReal ‖F n a - f a‖ ≤ ENNReal.ofReal (‖F n a‖ + ‖f a‖)
case hp
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 : HasFiniteIntegral bound
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)
n : ℕ
a : α
h₁ : ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
h₂ : ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)
⊢ 0 ≤ ‖F n a‖
case hq
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 : HasFiniteIntegral bound
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)
n : ℕ
a : α
h₁ : ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
h₂ : ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)
⊢ 0 ≤ ‖f a‖

Next tactic:
apply ofReal_le_ofReal