Denote by $|A|$ the measure of $A$ (Can be Lebesgue measure) under what conditions on a function $f:\mathbb{R}^m \to \mathbb{R}$ the preimage of a null set is zero. i.e.


$|A|=0 \Rightarrow |f^{-1}(A)| =0$


A special interest for conditions on not necessarily smooth functions



Preimage of zero measure sets
For example, it is sufficient that $f\in C^1$ and the set $\lbrace x | \nabla f(x) = 0 \rbrace$ has measure zero.


To prove this, note that this is true locally, in a neighborhood of each point where $\nabla f \neq 0$, due to the implicit function theorem.
Now the claim follows from the fact that $f^{-1}[A] \subset Z \cup \bigcup\_n (U\_n \cap f^{-1}[A])$, where $Z = \lbrace \nabla f = 0 \rbrace$ and $(U\_n)$ is a countable covering of $\mathbb{R}^m \setminus Z$ by neighborhoods for which the implicit function theorem applies.


