∀ {β : Type u_2} {E : Type u_4} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace E] [inst_2 : NormedAddCommGroup E]
  [inst_3 : BorelSpace E] {f : β → E} {μ : MeasureTheory.Measure β} (fmeas : Measurable f)
  [inst_4 : TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})],
  MeasureTheory.Integrable f μ →
    ∀ (n : ℕ), MeasureTheory.Integrable (⇑(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n)) μ