Let $A\subseteq R^n$ be a subset which is homeomorphic to $\left[0,1\right]^n$. Does there exists a homeomorphic map $f:A\rightarrow \left[0,1\right]^n$ such that for any $X_1,X_2\subseteq\left[0,1\right]^n$ with $\mu(X_1)=\mu(X_2)$, $\mu(f^{-1}(X_1))=\mu(f^{-1}(X_2))$ holds? $\mu$ is the Lebesgue measure.
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$\begingroup$ Up to rescale we can suppose that $A$ has measure $1$, and then the condition means that $f$ is measure-preserving. $\endgroup$– YCorCommented Apr 24 at 6:27
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4$\begingroup$ I am afraid that the answer to the question is negative for $n=2$ because the boundary of $A$ can have positive Lebesgue measure in the plane. $\endgroup$– Taras BanakhCommented Apr 24 at 6:31
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$\begingroup$ Examples for the comment of Taras Banakh are Oosgod curves (combined with the Schoenflies theorem). $\endgroup$– Jochen WengenrothCommented Apr 24 at 7:18
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$\begingroup$ More on Osgood curves here. $\endgroup$– KP HartCommented Apr 24 at 10:40
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$\begingroup$ However this is true for open $A$ and open cube. $\endgroup$– Alexandre EremenkoCommented Apr 24 at 11:38
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