Originally Posted by

**zebra2147** I would like to know how to do this problem before my test tomorrow and I'm not really sure how to do it. Any guidance would be appreciated.

Let $\displaystyle C$ be the Cantor set. Let $\displaystyle f:[0,1]\rightarrow \mathbb{R}$ be determined by $\displaystyle \[

f(x)=

\begin{cases}

1, &\text{when $x\in C $}\\

0, &\text{when $x\notin C $}\\

\end{cases}

\]$

Show that $\displaystyle f$ is Riemann integrable on $\displaystyle [0,1]$ and $\displaystyle \int _{0}^{1}f=0$. [That is the "length of the Cantor set is 0.]

I want to approach it like this but I'm not sure if it is correct:

We want to show that $\displaystyle f$ is integrable we need to show that $\displaystyle \{\sum S|S \text {is a lower step function of f}\} - \{\sum s|s \text {is a lower step function of f}\}<\epsilon$ for some $\displaystyle \epsilon >0$.

We can make $\displaystyle \sum s=0$ when $\displaystyle x\notin C$. So i think all we have to do is to show that $\displaystyle \sum S< \epsilon$. However, I'm not sure how to do this.