The indicator of $\QQ\cap[0,1]$ is Lebesgue integrable but not Riemann integrable

Let \(f\coloneqq\chi_{\QQ\cap[0,1]}\colon[0,1]\to\RR\), the Dirichlet function on \([0,1]\).

• The set \(\QQ\cap[0,1]\) is countable, hence has Lebesgue measure \(0\), so \(f\) is Lebesgue measurable, \(f = 0\) almost everywhere, and \(\int_{[0,1]} f\,dm = 0\); in particular \(f\) is integrable.

• Every subinterval of \([0,1]\) of positive length contains both rational and irrational points, so for every partition \(P\) of \([0,1]\) the upper Darboux sum is \(U(f,P) = 1\) and the lower Darboux sum is \(L(f,P) = 0\). Hence \(f\) is not Riemann integrable.