Example about a simple function not step function

Question

I am struggling with understanding that not every simple function is a step function. I am struggling with understanding that the characteristic function on the Cantor set is simple but not step function. Could you please explain what makes it not simple, I tied to show there is no partition of [0,1] such that $\chi_{_{\mathbf{C}}}$ is constant on the intervals in the partition but I could not.

One more thing, I would like another example of a simple function that is not a step function.

Thank you.

Answer 1

A step function takes a finite number of values, each on some interval. If the characteristic function of Cantor set is a step function it must take the value $1$ on a finite number disjoint of intervals which makes the Cantor set a finite union of intervals. Since Cantor set has measure $0$ and no (non-degenerate) interval has measure $0$ this is a contradiction. Another example of s simple function which is not step function is characteristic function of the set of all rational numbers.