Measure zero sets to measure zero sets

Asked Feb 14 '16 at 12:26

Active Feb 14 '16 at 12:43

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I would like to collect functions which preserves measure zero sets..

Let $f:A\subseteq \mathbb{R}\rightarrow \mathbb{R}$ is a function such that $f(A)$ is of measure zero if $A$ is of measure zero..

I know some examples..

Lipschitz continuous functions
Absolutely continuous functions

I want to make this community wiki..

Please give only one type of function in one answer..

EDIT : This is a question that asks to collect all (not highly non trivial) functions that satisfies some property.. The question that has is linked to this question does not solve the problem as that is saying about some specific case of topological vector spaces and some hausdorff measure.. That is about some particular problem but here i want to collect all functions that satify the condition above...

edited Feb 14 '16 at 12:43

asked Feb 14 '16 at 12:26

kk lm

6

This is exactly Lusin's N condition. See for instance here or here. – Watson Feb 14 '16 at 12:30
related – Pedro Feb 14 '16 at 12:36
@Pedro : I am trying to collect all such functions.. Please write it as an answer (in your words if possible)... – kk lm Feb 14 '16 at 12:38
@Watson : I am trying to collect all such functions.. The question that you have linked is some special case.. I am looking for collection of all functions... – kk lm Feb 14 '16 at 12:48
The Encyclopedia of Mathematics shows some examples. – Pedro Feb 14 '16 at 13:11
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@Pedro : Thanks.. That is very mych related – kk lm Feb 14 '16 at 13:26
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I believe this Question could be improved if the OP would invest the effort to understand the general result linked to as a duplicate, and then make adjustments to narrow the current problem. The specification "not highly non trivial" could in particular stand some clarification. – hardmath Feb 14 '16 at 13:42

Measure zero sets to measure zero sets

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