Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound 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)
⊢ ∀ (n : ℕ), AEMeasurable (fun a ↦ ENNReal.ofReal ‖F n a - f a‖) μ

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound 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 ≠ ∞

pp: case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound 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)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)

Tactic:
· exact fun n =>
      measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound 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 ≠ ∞

case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound 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)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)