# 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.

• It seems like you are looking for Lusin N-property. Commented Sep 14, 2012 at 10:25
• @Matt The cardinality argument argument tells you nothing and is in fact wrong. See here for a counterexample (the image of the Cantor set under the Cantor-Lebesgue function has measure one) and here for a proof of the statement in the question.
– t.b.
Commented Sep 14, 2012 at 10:39
• @nikita2: That property references the real line and Lebesgue measure. I'm basically asking for sufficient conditions for a generalization of the Lusin N property. Commented Sep 14, 2012 at 11:12
• There are results for Sobolev maps (more general than the Lipschitz maps), but here the metric structure is even more involved than in the Lipschitz case. I never saw a nontrivial condition for property N which did not involve a metric. A trivial non-metric condition would be: all subsets of $Y$ with positive measure have larger cardinality than any subset of $X$ with zero measure.
– user31373
Commented Sep 14, 2012 at 15:31
• Of course the minimal condition can be formulated quite easily: It is "the function has the property that it maps null sets to null sets." :-) Commented Sep 15, 2012 at 14:19

## 1 Answer

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.