19

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.

Community
  • 1
Ricardo Buring
  • 6,649
  • 7
  • 35
  • 63
  • Regarding your first sentence: is there anything to prove? If $f$ is a function, the cardinality of the image will be less equals the cardinality of $A$, so will have smaller equal Lebesgue measure. In particular: why do you require $f$ to be Lipschitz? – Rudy the Reindeer Sep 14 '12 at 10:17
  • 1
    It seems like you are looking for Lusin N-property. – Nikita Evseev Sep 14 '12 at 10:25
  • 3
    @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. Sep 14 '12 at 10:39
  • Matt: Let $C$ be the Cantor set and $f$ the function that maps $C$ onto $[0,1]$. Isn't that a counterexample to your claim? – Ricardo Buring Sep 14 '12 at 10:41
  • @Matt: In $\mathbb R^2$, the cardinality of the unit disk (Lebesgue measure $\pi$) and the cardinality of a straight line (Lebesgue measure $0$) is the same. Thus there exists a surjective function from the straight line to the unit disk. Of course you can extend that function to all of $\mathbb R^2$. – celtschk Sep 14 '12 at 10:42
  • @celtschk: and you can make it continuous by taking a Peano curve and using Tietze's extension theorem... – t.b. Sep 14 '12 at 10:44
  • 1
    @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. – Ricardo Buring Sep 14 '12 at 11:12
  • 1
    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. –  Sep 14 '12 at 15:31
  • @t.b. Thank you lots for the comment. I thought it might be wrong and hoped someone would tell me if I commented. I will read the links, but right now I'm too tired. – Rudy the Reindeer Sep 14 '12 at 16:17
  • 5
    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." :-) – celtschk Sep 15 '12 at 14:19
  • 1
    I don't understand how you can be looking for a condition which doesn't involve the metric structure. If the two measures are arbitrary rather than depending on the metric space (eg Hausdorff measure), then nothing about the map itself can tell you what sets will be nullsets in the image measure space (for instance, take some measure on the image space, and add a measure supported on the image of some nullset). – jwg Dec 22 '13 at 09:00
  • @jwg Good point (it seems rather obvious now); if you post your comment as an answer I will accept it. – Ricardo Buring Dec 22 '13 at 14:43

1 Answers1

3

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.

jwg
  • 2,846