# Expanding Measurable Sets

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 $\mu(S) \leq \mu(T)$?

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@Alex, you need $T$ to be measurable; otherwise you can not write the inequality. – Anton Petrunin Jun 10 '11 at 13:49