>>11555197Riemann integral is very simple to understand, but the definition doesn't say much and you need to stabilish a lot from the ground up.
Labesgue is more sophisticated (usually founded on the idea of measure), but easier to work with in theorems.
It also agrees with the Riemann integral in every case where the Riemann integral is well defined, what means theorems about Lebesgue integral translate to the Riemann integral.
>>11555255The classic example is Dirichlet's function (the indicator function of the rationals). There's a bunch of other weird functions people made that also aren't Riemann integrable but are Lebesgue integrable.
I vaguely remember a professor mentioning that non-Riemann integrable functions come up in Ergodic theory, but I know jack about the Ergodic theory so I don't have any specific example from there.
