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Comment by mturmon | Hacker News Reader

To admit delta functions (e.g., impulse responses) you need at least a Riemann-Stieltjes integral. And that is (as far as I know) something of a patch.

Conventional Hilbert spaces (e.g., for QM) are defined with respect to the Lebesgue integral. If you restrict to just Riemann-integrable functions, the Hilbert space isn't complete (IIRC) because not enough functions qualify.

Besides all this, you can argue that the Lebesgue integral is necessary to physics because it is the best known foundation for probability theory, which (again, ultimately, due to the observable manifestations of QM) is central to our understanding of the physical world. E.g., ergodic theory, Brownian motion, magnetism, phase transitions, and on and on.

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Well, Lebesgue integration cannot handle delta functions either since they are not even functions but distributions.

(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.)

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.

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.

IIRC, a Hilbert space cannot contain the Dirac delta function regardless of the integral used, since the inner product of the delta function with itself is ill-defined. So what people use in QM is technically "rigged Hilbert spaces" (or Gelfand triples) which a few mathematicians go to great pains to formalise and put on a rigorous foundations. But about which exactly zero shits are given by all the physics departments of the world; they simply handwave about with some box regularizations, swap integrals and limits at their hearts desire, and everything turns out fine in the end.

Yes, I wasn't meaning to imply delta "functions" are elements of Hilbert spaces. My first paragraph was just about delta functions as tools for representing impulse responses for mechanical or electrical systems, not QM.

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