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Say I have a homeomorphism $f:A\longrightarrow B$ between open subsets $A$ and $B$ of $\mathbb{R}^n$. If $S\subset A$ has positive Lebesgue measure, does $f(S)$ also have positive measure? If so, do you know a reference to this result?

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    $\begingroup$ For example: actamath.savbb.sk/pdf/acta1906.pdf $\endgroup$ – Alex R. Jun 20 '16 at 21:23
  • $\begingroup$ I made a mistake, I meant an image of a positive measure set $S$, not an open set $A$. $\endgroup$ – Tom Jun 20 '16 at 21:38
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No, the image of a set of positive measure under a homeomorphism can have measure zero.

Let $\phi:[0, 1] \to [0, 1]$ be the Cantor function, a.k.a. the Devil's staircase. Define $g:[0, 1] \to [0, 2]$ by $g(x) = x + \phi(x)$, and let $f = g^{-1}:[0, 2] \to [0, 1]$.

Since $g$ is continuous and strictly increasing, it is a bijection, hence a homeomorphism because a closed interval is a compact Hausdorff space.

The image of the Cantor ternary set under $\phi$ is $[0, 1]$, so the image $S$ of the Cantor set under $g$ has positive measure (in fact, measure $1$). Consequently, the image of $S$ under the homeomorphism $f$ is the Cantor ternary set, which has measure zero.

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