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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!

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However, I am not sure how to do the computation for lower and upper riemann integrals nor how to conclude b.

To do the computation: For the upper integral, integrate the supremum. For the lower integral, integrate the infimum.

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 $x$?

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  • $\begingroup$ what do you mean integrate the supremum? Am I not supposed to take the sup/inf over all partitions to get the lower/upper riemann integral? And if so, I am not even sure how to assert the existence of this integral $\endgroup$ – Kaan Yolsever Mar 13 at 18:13
  • $\begingroup$ The formal definition of the integral is not generally a good way to actually compute the integral - but here it doesn't matter. Every partition has exactly the same upper sum and exactly the same lower sum. $\endgroup$ – jmerry Mar 13 at 18:23
  • $\begingroup$ I really don't know how to compute the lower and upper riemann integrals informally. Can I just take the integral of it? Isn't that assuming the integral exists which we are trying to prove in this case $\endgroup$ – Kaan Yolsever Mar 13 at 18:32
  • $\begingroup$ 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. $\endgroup$ – jmerry Mar 13 at 18:39
  • $\begingroup$ 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 $\endgroup$ – Kaan Yolsever Mar 13 at 18:44

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