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Limit of uniformly converging volume-preserving homeomorphisms

Definition A continuous map $f\colon \mathbb{R}^n \to \mathbb{R}^n$ is volume-preserving if, for every Borel set $V\subset\mathbb{R}^n$, $\mathcal{L}^n(V) = \mathcal{L}^n(f^{-1}(V))$. I am wondering ...