I am getting pretty frustrated with the Lebesgue integral mainly because it seems highly impractical to calculate anything non-trivial. Whenever I look for a concrete calculation all I see are encomiums about how wonderful it is and then invariably the only concrete calculation is the Dirichlet function where unsurprisingly the measures are easy to calculate. When a run of the mill function is offered to be calculated ,I have seen one of two kinds of responses:
1) The answer involves a trick that can't be generalized
2) The answer given is " The function is Riemann integrable so just use that" i.e forget about the Lebesgue integral.
The only method I have seen that aspires to practicality is to use the monotone convergence theorem i.e get a bunch of simple functions whose limit is the function you want to integrate. Integrate them and take the limit. I have tried this for $x^2$ and I run into hard sums which are summed by guess what...help of the Riemann integral.(could be that I chose an inconvenient set of simple functions but that partly proves my point-very easy to make things complicated)
So is the Lebesgue integral mostly used in formal situations and then occasionally some highly pathological function is pulled to justify the work? Are there examples where the Lebesgue integral is of practical importance and there can be no recourse to the Riemann integral? Highly discontinuous functions are not welcome.