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?

  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.