On the uniform strong Lusin condition

Assume that f is a function defined on some interval I ? ? ^m . Literature offers several equivalences of the type: f has a property P (like absolute continuity, bounded variation, etc.) on I if and only if f has P on each (closed) null subset of I. Such results feature a pretty important role in the integration theory. We make a brief review of these results and then provide an example showing that they can break down if considered in the uniform version, that is, for sequences of functions instead of a particular function.