Riemann Sum

**chabmgph** · December 13th 2008, 01:01 PM

Hi,

I am drawing a blank here.

I know it has to do with the fact that the rationals are dense in the reals, but somehow it is not coming together for me. Any idea?

Let $f : [ 0,1] \times [0,1] \rightarrow \mathbb{R}$ be defined by $f(x, y)=\begin{Bmatrix} 1 & x\text{ rational} \\ 2y & x \text{ irrational.} \end{Bmatrix}$
Show that the iterated integral $\int_0^1 [\int_0^1 f(x,y)dy]dx$ exists but that $f$ is not integrable.

I am supposed to use the following to prove the part where it is not integrable:

Suppose $f: R \rightarrow \mathbb{R}$ is bounded form the sum $S_n=\sum_{jk=0}^{n-1} f(C_{jk})\triangle x \triangle y = \sum_{jk=0}^{n-1} f(C_{jk}) \triangle A$ .
If the sequence $(S_n)_n$ converges to a limit $S$ as $n \rightarrow \infty$ and if the limit $S$ is the same for any choice of sample points $C_{jk}$ in $R_{jk}$ , then we say that $f$ is integrable over $R$ .

In other words, I can't use the uppersum not equal to the lower sum strategy since it is not cover in class which I am taking now.

Any help is appreciated. Thanks for your time!

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