# Computation of Riemann integral

Let $$f:[0,1]$$ $$\times$$ $$[0,1]$$ $$\rightarrow$$ $$\mathbb{R}$$ where $$\begin{array} f(x,y) = \begin{cases} 1 & \text{if x \in \mathbb{Q}} \\ 2y &\text{if x \notin \mathbb{Q}} \end{cases} \end{array}$$

Compute the Upper and Lower Riemann Integrals \begin{align} \overline{\int_{0}^{1}}f(x,y)dx && \text{and} && \underline{\int_{0}^{1}}f(x,y)dx \end{align} in terms of y and show that \begin{align*} \int_{0}^{1}f(x,y)dy \end{align*} exists for each fixed x.

$$\textbf{Attempt:}$$

if $$y < 1/2$$ since rationals are dense in irrational and vice versa, we know the infimum for the indicator function is 2y in any given interval for any partition and supremum is 1. If $$y > 1/2$$ then supremum is 2y and infimum is 1 for any given interval. However, I am not sure how to do the computation for lower and upper riemann integrals nor how to conclude the second part.

Thank you!

For the second part, note that it's talking about integrating with respect to the other variable. What is $$f(x,y)$$ for some fixed $$x$$?
• The upper and lower integrals always exist. The integral as a whole only exists if they're equal to each other - which, for the $dx$ integral here, isn't the case. – jmerry Mar 13 at 18:39
• How can we know that $sup_P{\sum_i M_i \delta(x_i)$ always exists? Could you write out what you are claiming? I am having a hard time following – Kaan Yolsever Mar 13 at 18:44