I am stuck in a simple question which should be easy.
The indicator function of $\mathbb Q$ on the compact interval $[0,1]$ is not Riemann integrable. The proof is very easily done by taking the upper and lower sums which are not equal. Yet I cannot understand why the Lebesgue inegrability condition does not seem to work: a bounded function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its points of discontinuity form a set with Lebesgue measure zero.
So the question is: does the set od discontinuities for the indicator function of $ \mathbb Q$ on $[0,1]$ have Lebesgue measure zero? The problem arose when I was reading a book which stated that the indicator function is continuous on the set of the irrationals, which itself is un uncountable set! So what is wrong?