Practicality of the Lebesgue integral

Question

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.

Answer 1

Andrew D Lewis wrote a nice essay giving what he considers to be "The correct defence of the Lebesgue integral." Here is the Abstract:

It is well-known that the Lebesgue integral generalises the Riemann integral. However, as is also well-known but less frequently well-explained, this generalisation alone is not the reason why the Lebesgue integral is important and needs to be a part of the arsenal of any mathematician, pure or applied. Those who understand the correct reasons for the importance of the Lebesgue integral realise there are at least two crucial differences between the Riemann and Lebesgue theories. One is the difference between the Dominated Convergence Theorem in the two theories, and another is the completeness of the normed vector spaces of integrable functions. Here topological interpretations are provided for the differences in the Dominated Convergence Theorems, and explicit counterexamples are given which illustrate the deficiencies of the Riemann integral. Also illustrated are the deleterious consequences of the defects in the Riemann integral on Fourier transform theory if one restricts to Riemann integrable functions.

Answer 2

You can very well live inside the world of the Riemann integral. As far as the dominated convergence theorem goes, there is Arzela's theorem that will do almost exactly the same: It allows you to exchange the limit operation with integration as long as the functions are bounded. However, the repeated applications of limits is where the Riemann theory fails. This is why you need the Lebesgue, Daniell, or Henstock–Kurzweil integral. As you can see, there is more than just one way to make this generalization, or to skin the proverbial cat. The Lebesgue theory was simply there first.