In "practice" it is likely "useless", because the functions that it can integrate that Riemann can't are just too funky to appear in practice :-)

Its value is much more indirect and subtle, as in "it serves as a solid basis of <quantum mechanics>" due to the definition of <Hilbert spaces>.

Bibliography:
* https://math.stackexchange.com/questions/53121/how-do-people-apply-the-lebesgue-integration-theory
* https://www.quora.com/What-are-some-real-life-applications-of-Lebesgue-Integration


Real world applications of the Lebesgue integral