what functions or classes of functions are Riemann non-integrable but Lebesgue integrable I am wondering if there are some other examples of Riemann non-integrable but Lebesgue integrable, besides the well-known Dirichlet function.
Thanks.
 A: Yuval's link shows the general criterion for bounded functions on a bounded interval, namely continuity almost everywhere.  The Dirichlet function is a good example, but it is almost everywhere zero, and the zero function sure is Riemann integrable.  If you consider the characteristic function of a Cantor set of positive measure, then it is not Riemann integrable because it is discontinuous at each point in this "fat" Cantor set, and you cannot modify the values on a set of measure 0 to fix this defect.
The same idea would work for any closed nowhere dense subset $E\subset[0,1]$ of positive measure, and you can appeal to the definition to see this.  Because every subinterval of $[0,1]$ has nonempty intersection with $[0,1]\setminus E$, the lower Riemann sums of $\chi_E$ are all $0$.  However, the upper Riemann sums are all bounded below by $m(E)\gt0$.
A: 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).
A: Check the Wikipedia article on the Riemann integral for some ideas.
