∀ {α : Type u_1} {ι : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι}
  [inst_1 : NormedAddCommGroup E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α},
  MeasureTheory.AECover μ l φ →
    ∀ {f : α → E} (I : ℝ),
      (∀ (i : ι), MeasureTheory.IntegrableOn f (φ i) μ) →
        (∀ᶠ (i : ι) in l, ∫ (x : α) in φ i, ‖f x‖ ∂μ ≤ I) → MeasureTheory.Integrable f μ