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Let $(X, \mathcal{M}, \mu)$ be a measure space. Is there a name for this property for a function $f: X \to X$?

For all $\epsilon > 0$, there exists a $\delta > 0$ such that for all $E \in \mathcal{M}$, if $\mu(E) < \delta$, then $\mu(f(E)) < \epsilon$.

More specifically, how is this property related (via necessary and/or sufficient conditions) (esp. in the case $X = \mathbb{R}$) to other well-studied properties such as absolute continuity, measurability, etc.?

More details when $X = \mathbb{R}$:

  • An $f$ satisfying this property need not be continuous.

  • I believe that if $f$ is absolutely continuous, then this property holds.

  • On the other hand, even if I assume $f$ is continuous, I am not sure it's true that this property implies absolutely continuous, due to some problems with overlap.

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Yes, I think this property is called absolute continuity. There are functions which are continuius but not absolutely continuous. For instance, the Cantor function mapping a Cantor set if measure zero into the unit interval. Or the Peano curve mapping unit interval into the unit square. Furthermore, both functions do not satisfy Luzin's N property in the sense that they map zero measure sets onto sets of positive measure.

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  • $\begingroup$ Well, I guess you are saying that a continuous function with this property is absolutely continuous? Certainly there are functions which satisfy this property and are not absolutely continuous (just take any discontinuous function satisfying this property.) $\endgroup$ – 6005 Nov 27 '13 at 22:33

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