Denote by $\displaystyle m$ the Lebesgue measure on $\displaystyle X = (0, 1)$.

1. Prove that the Dirichlet function $\displaystyle d(\cdot)$ is not Riemann integrable.

Since the Dirichlet function is discontinuous at every point $\displaystyle x \in \mathbb{R}$ it is not Riemann integrable by definition.2. Prove that $\displaystyle d(\cdot)$ is a non-negative Lebesgue measurable simple function.

3. What is the Lebesgue integral $\displaystyle \int d(x)dm$?

I think it is zero but I don't know how to show it.4. Prove that the Riemann function $\displaystyle r(\cdot)$ is Riemann integrable.