# Does the image of positive measure set under homeomorphism also have positive measure?

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?

• For example: actamath.savbb.sk/pdf/acta1906.pdf – Alex R. Jun 20 '16 at 21:23
• I made a mistake, I meant an image of a positive measure set $S$, not an open set $A$. – Tom Jun 20 '16 at 21:38

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.