
Riemann Sum
Hi,
I am drawing a blank here. (Doh) 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? (Thinking)
Let $\displaystyle f : [ 0,1] \times [0,1] \rightarrow \mathbb{R}$ be defined by $\displaystyle f(x, y)=\begin{Bmatrix} 1 & x\text{ rational} \\ 2y & x \text{ irrational.} \end{Bmatrix} $
Show that the iterated integral $\displaystyle \int_0^1 [\int_0^1 f(x,y)dy]dx$ exists but that $\displaystyle f$ is not integrable.
I am supposed to use the following to prove the part where it is not integrable:
Suppose $\displaystyle f: R \rightarrow \mathbb{R}$ is bounded form the sum $\displaystyle S_n=\sum_{jk=0}^{n1} f(C_{jk})\triangle x \triangle y = \sum_{jk=0}^{n1} f(C_{jk}) \triangle A$.
If the sequence $\displaystyle (S_n)_n$ converges to a limit $\displaystyle S$ as $\displaystyle n \rightarrow \infty$ and if the limit $\displaystyle S$ is the same for any choice of sample points $\displaystyle C_{jk}$ in $\displaystyle R_{jk}$, then we say that $\displaystyle f$ is integrable over $\displaystyle R$.
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. (Doh)
Any help is appreciated. Thanks for your time!