When is the image of a null set also null? It is easy to prove that if $A \subset \mathbb{R}$ is null (has measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz then $f(A)$ is null. You can generalize this to $\mathbb{R}^n$ without difficulty.
Given a function $f: X \rightarrow Y$ between measure spaces, what are the minimal conditions (or additional structure) needed on $X$, $Y$ and $f$ for the image of a null set to be null?
Any generalization (containing the above as a special case) is appreciated. Apparently if $X$ and $Y$ are $\sigma$-compact metric spaces with the $d$-dimensional Hausdorff measure and $f$ is locally Lipschitz then the result holds. Can we be more general? I would like to see something without a metric.

There is no real condition on a map which could be valid for arbitrary measure spaces. If the measures can be arbitrary, rather than depending on the metric space structure of the underlying space, (eg Hausdorff measure), then nothing about the map itself can tell you which sets will be nullsets in the image measure space.
For example, given some measure on the image space and a map from some other measure space which sends nullsets to nullsets, add a measure supported on the image of some nullset. The map no longer maps nullsets to nullsets.
