In my limited experience, I've only seen Lebesgue integration theory used to prove things that rely at the base on a normal integral using anti-differentiation, which is thus indistinguishable from the process of Riemann integration.
Does anyone know of an example of actually integrating a non-Riemann-integrable function using Lebesgue integration, using the actual limit defn of the integral?
EDIT: Looking for one with an integral that is not 0 or 1 :)
The function $$f: [0,1] \to \mathbb R, f(x) = \begin{cases} 1, \text{ if $x \in \mathbb Q$} \\ 0, \text{ if $x \notin \mathbb Q$} \end{cases}$$
is not Riemann intégrable (upper sum $1$, lower sum $0$). But Lebesgue intégrable with integral $0$ obviously.