[Math] Example for non-rectifiable set

Question

The definition of rectifiable set is that if the constant function $1$ is integrable over the set $S$, then the set $S$ is rectifiable. What can be an example for non-rectifiable sets? I cannot visualize it.

Thanks for your help!

Answer 1

For an example of a non-rectifiable curve, we may take any space-filling curve, i.e. a continuous surjective map $[0,1]\to [0,1]\times [0,1]$. This is indeed difficult to visualize and was one of those things that 19th century mathematicians were upset to discover existed. In the language given in this Wikipedia article, $[0,1]\times [0,1]$ is not a $1$-rectifiable set.