TITLE: Maps that preserve measure zero property
QUESTION [7 upvotes]: Consider a map $f: \mathbb{R}^n \to \mathbb{R}^m$ that is differentiable (usually even smooth). If $B \subset \mathbb{R}^m$ has measure zero (Lebesgue measure), then what types of maps $f$ satisfy $A = f^{-1}(B)$ also has measure zero?
To provide some context: I have a property $\mathcal{P}$ that holds almost everywhere in $\mathbb{R}^m$; now I want to characterize the class of maps $f$ such that $\mathcal{P}(f(\cdot))$ holds almost everywhere in $\mathbb{R}^n$

REPLY [2 votes]: I was searching for a solution for the problem that you are working on. I found an interesting paper about it:
Ponomarev, S. P. - Submersions and preimages of sets of measure zero.pdf
It says that, if a function is "sufficiently" smooth and the rank of the Jacobian matrix satisfy some properties, then the preimage has also measure zero.
Best,