= Real world applications of the Lebesgue integral
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