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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 $\mathbb{R}^n$

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    @studiosus You are correct. I was in a review queue, so I had limited opportunity or motivation to verify the correctness of the assertion beyond that it didn't answer the question.2013-10-15

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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,