Suppose $f:\mathbb{R}^n\to\mathbb{R}^n$ is an injection, such that for any $E\subseteq\mathbb{R}^n$ we have $m^*(E)=m^*(f(E))$. Is it true that for any $E\subseteq\mathbb{R}^n$ measurable, $f(E)$ is always measurable?
2026-04-03 14:23:49.1775226229
Do injective outer measure-preserving functions preserve measurable sets?
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