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)?