Denote by the Lebesgue measure on .

is the Dirichlet function.

Consider the sequence of functions on defined as follows:

1) Prove that , and that .

2) Are the functions Riemann integrable? If your answer is yes, what is ?

3) Are the functions Lebesgue integrable? If your answer is yes, what is the Lebesgue integral ?

4) Is it true that (the Riemann integral)?

5). Is it true that (the Lebesgue integral)?