-- Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence
Goal:
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hf_tendsto : Tendsto (fun (i : ℕ) ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone (β := ℝ≥0∞) fun (i : ℕ) ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
h_exists : ∀ᵐ (a : α) ∂μ, ∃ (l : ℝ≥0∞), Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l)
let let₁ let₂ : ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F a))
let₃ :
  let F' : α → ℝ≥0∞ := fun (a : α) ↦
    if h : ∃ (l : ℝ≥0∞), Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l) then
      h.choose (p := fun (l : ℝ≥0∞) ↦ Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l))
    else ∞;
  ∀ (hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F' a))),
    ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F a))
let₄ :
  let F' : α → ℝ≥0∞ := fun (a : α) ↦
    if h : ∃ (l : ℝ≥0∞), Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l) then
      h.choose (p := fun (l : ℝ≥0∞) ↦ Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l))
    else ∞;
  ∀ (hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F' a))), F' =ᶠ[ae μ] F
let₅ :
  let F' : α → ℝ≥0∞ := fun (a : α) ↦
    if h : ∃ (l : ℝ≥0∞), Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l) then
      h.choose (p := fun (l : ℝ≥0∞) ↦ Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l))
    else ∞;
  ∀ (hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F' a))) (hF'_le : F ≤ᶠ[ae μ] F'),
    F' =ᶠ[ae μ] F
let₆ :
  let F' : α → ℝ≥0∞ := fun (a : α) ↦
    if h : ∃ (l : ℝ≥0∞), Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l) then
      h.choose (p := fun (l : ℝ≥0∞) ↦ Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l))
    else ∞;
  ∀ (hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F' a))) (hF'_le : F ≤ᶠ[ae μ] F'),
    ∫⁻ (a : α), F' a ∂μ = ∫⁻ (a : α), F a ∂μ
let₇ :
  let F' : α → ℝ≥0∞ := fun (a : α) ↦
    if h : ∃ (l : ℝ≥0∞), Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l) then
      h.choose (p := fun (l : ℝ≥0∞) ↦ Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 l))
    else ∞;
  ∀ (hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F' a))) (hF'_le : F ≤ᶠ[ae μ] F'),
    Tendsto (fun (i : ℕ) ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun (i : ℕ) ↦ f i a) atTop (𝓝 (F a))

Tactic:
classical
  let F' : α → ℝ≥0∞ := fun a ↦ if h : ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l)
    then h.choose else ∞
  have hF'_tendsto : ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F' a)) := by
    filter_upwards [h_exists] with a ha
    simp_rw [F', dif_pos ha]
    exact ha.choose_spec
  suffices F' =ᵐ[μ] F by
    filter_upwards [this, hF'_tendsto] with a h_eq h_tendsto using h_eq ▸ h_tendsto
  have hF'_le : F ≤ᵐ[μ] F' := by
    filter_upwards [h_bound, hF'_tendsto] with a h_le h_tendsto
    exact ge_of_tendsto' h_tendsto (fun m ↦ h_le _)
  suffices ∫⁻ a, F' a ∂μ = ∫⁻ a, F a ∂μ by
    refine (ae_eq_of_ae_le_of_lintegral_le hF'_le h_int_finite ?_ this.le).symm
    exact ENNReal.aemeasurable_of_tendsto hf_meas hF'_tendsto
  refine tendsto_nhds_unique ?_ hf_tendsto
  exact lintegral_tendsto_of_tendsto_of_antitone hf_meas hf_mono h0 hF'_tendsto

Premises used: tendsto_nhds_unique