1. ## Analysis, Dirichlet function & Riemann function

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.

2. For 2, ask yourself this: how might you define the Dirichlet function as the characteristic function of a set?

For 3, I’m not quite sure what you mean by $\displaystyle f_N(x)$.

For 4, I am not sure if you know Lebesgue’s integrability criterion (this is a criterion for Riemann integrability due to Lebesgue, not a condition for Lebesgue integrability). Basically, a function is Riemann integrable iff it is continuous $\displaystyle m$-almost everywhere on the domain. So a straightforward way to show this is to note that the number of discontinuities of $\displaystyle r$ is countable.

3. Originally Posted by Tikoloshe
For 3, I’m not quite sure what you mean by $\displaystyle f_N(x)$.
I fixed it. It should be $\displaystyle d(x)$, not $\displaystyle f_N(x)$

4. So what about the rest? Once you know 2, the answer of 3 follows as the measure of the set whose characteristic function you consider. If you are still having difficulty, try this:

Write the definition of a characteristic function $\displaystyle \chi_E(x)$ for some measurable set $\displaystyle E$. Then write the definition of $\displaystyle d(x)$. Compare the two forms, and see if you find a set whose characteristic function is $\displaystyle d$.