Well, Lebesgue integration cannot handle delta functions either since they are not even functions but distributions.<p>(Incidentally, there is such a thing as Lebesgue-Stieltjes integral, the kind of issues Riemann-Stieltjes solve are orthogonal to the issues that motivate Lebesgue integration.)<p>As for the foundations of probability theory I am skeptical. I agree the theorems are nicer if you base probability theory on the Lebesgue integral, but I very much doubt if there is a realistic physical question that could not be answered by probability theory based on the Riemann integral.<p>And for what it is worth, a lot of modern physics (i.e., all of quantum field theory and modern statistical physics) is based on a type of integral (the path integral) that does not yet have rigorous mathematical foundations.