Denote by the Lebesgue measure on .

1. Prove that the Dirichlet function is not Riemann integrable.

Since the Dirichlet function is discontinuous at every point it is not Riemann integrable by definition.2. Prove that is a non-negative Lebesgue measurable simple function.

3. What is the Lebesgue integral ?

I think it is zero but I don't know how to show it.4. Prove that the Riemann function is Riemann integrable.