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.<p>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&#x27;t complete (IIRC) because not enough functions qualify.<p>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.