Let $\Omega$ be an open subset of $\mathbf{R}^n$. For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ is of zero one-dimensional Hausdorff measure zero. Note that f is not assumed to be a homeomorphism.

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    $\begingroup$ What is $\Omega$ here? $\endgroup$ – Tom Leinster Dec 5 '12 at 15:02
  • $\begingroup$ Since $f: \Omega \to \mathbb{R}^N$, $f^{-1}:f(\Omega) \to \Omega$, so if $E \subset f(\Omega)$, we should not be measuring the 1-D lebesgue measure, but rather the 1-D Hausdorff measure $H^1$. $\endgroup$ – Daniel Spector Dec 5 '12 at 17:32
  • $\begingroup$ Yes, you are right. It is better to use the Hausdorff measure H^1. \Omega is a domain in \bR^n and f is not necessarily to be a homeomorphism here. $\endgroup$ – Changyu Guo Dec 6 '12 at 8:11
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    $\begingroup$ This is the worst question title in the history of Math Overflow. It provides literally no information about what the question is about. $\endgroup$ – arsmath Dec 6 '12 at 12:14
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    $\begingroup$ @arsmath after almost 7 years of hard thinking I changed to a more informative title $\endgroup$ – YCor Nov 15 '19 at 9:10

There may be a name for this, but it seems like a strange condition. Such a function cannot take a constant value on any set of positive Lebesgue measure, otherwise the inverse image of that constant (having zero 1-D Hausdorff measure in the range) would have positive Lebesgue measure, and therefore infinite 1-D Hausdorff measure.

A good start might be to investigate the situation on maps $f:[0,1] \to \mathbb{R}$ with the Lebesgue measure in both places.

There is also a related notion, called Lusin's N property, which means $f$ takes sets of measure zero into sets of measure zero (as opposed to $f^{-1}$, as you desire). This is a quality of Lipschitz functions that Sobolev functions also inherit, and is necessary to satisfy the fundamental theorem of Calculus (along with being differentiable a.e., etc.).

  • $\begingroup$ In the one-dimensional case suggested in this answer, the image measure (also called push-forward) $f_\ast\lambda (E) = \lambda(f^{-1}(E))$ is dominated by $\lambda$ hence, by Radon-Nikodym, it has a density. This may help to get something more concrete in particular cases. A problem with the higher dimensional case is that the one-dimensional Hausdorff measure is not $\sigma$-finite and Radon-Nikodym is not applicable. $\endgroup$ – Jochen Wengenroth Dec 7 '12 at 13:05

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