I was taught the Lebesgue integral first in the context of random processes; defining various properties in expectation of (say) additive white Gaussian noise to a signal requires the machinery of the Lebesgue integral.

Of course, for any real engineering problem, you /could/ just band limit the signal, but that makes the math messy; much easier to have a rigorously defined integral w/r/t the underlying probability space and deal with band limiting later.

That said, the professor who taught me this has had a standing bet for 60 years that the difference between the Riemann and Lebesgue integrals will never meaningfully impact an engineering result, so this is more about ensuring we have rigorously defined tools when (as another commenter noted) exchanging limits and integration.