Question

Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that $$ \|f(x)-f(y)\| \;\geq\; \|x-y\| $$ for all $x,y \in S$. Does it follow that $\mu(S) \leq \mu(T)$?

Answer 1

It follows from two observations:

• For Hausdorff meeasure your statement follows from the definition.
• Hausdorff measure = Lebesgue measure (up to constant).