apply MeasureTheory.Measure.integrableOn_of_bounded ?_ ?_ integrability_aux₀
a b : ℝ
⊢ MeasureTheory.IntegrableOn (fun x => ↑⌊x⌋) [[a, b]] MeasureTheory.volume
a b : ℝ
⊢ MeasureTheory.volume [[a, b]] ≠ ⊤

a b : ℝ
⊢ MeasureTheory.AEStronglyMeasurable (fun a => ↑⌊a⌋) MeasureTheory.volume