Ask Poincaré. The Lebesgue integral was so important, so relevant to his eyes that he never took the trouble to say anything about it between 1904 (at the latest) and 1912 nor to discuss it with Lebesgue or Borel, as far as we know. See An historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?
Edit: two downvotes, already. Please tell me why. If Poincaré does not need the Lebesgue integral (but only Riemann's or Cauchy's... According to Lebesgue himself, Poincaré was aware of his integral.), it means that it is nevertheless possible to do some good mathematics and physics without it, isn't it?