n dimensional measure of homeomorphism of interval

Question

Let $f: [a,b] \rightarrow I \subset \mathbb{R^n}$ (where $[a,b] \subset \mathbb{R}$ and $n>1$) be homeomorphism. Can the n-dimensional lebesgue measure of $I$ be positive?

Answer 1

Yes. What you're after is called an Osgood curve.

Here is a fractal example in dimension $2$, from Wikipedia. You start from a triangle (which could have dimension $n$ harmlessly) and at each step you remove a smaller triangle with thinner and thinner width.