TITLE: what functions or classes of functions are Riemann non-integrable but Lebesgue integrable
QUESTION [7 upvotes]: I am wondering if there are some other examples of Riemann non-integrable but Lebesgue integrable, besides the well-known Dirichlet function.
Thanks.

REPLY [3 votes]: For me the "canonical" example is the characteristic function of the rational numbers in $[0,1]$. The upper integral is one, the lower integral is zero.
Edit: Jonas and Moron have informed me that some people call this example Dirichlet function (and I vaguely remember that we might have done so as well in our Analysis class).