I understand that there are functions where the Lebesgue integral exists, but they are not Riemann integrable (e.g. the Dirichlet function). Are there also functions that are Riemann integrable but not Lebesgue integrable?
If we e.g. have a function $f(x)=x$, $x\in[0,5]$ and have to find the Lebesgue integral of that function $\int_x x d\lambda$ how is the calculus then. I know it should be the same as when we integrate with respect to Riemann (25/2). But how do we calculate it if we don't use the Riemann integral?