On the uniform strong Lusin condition

Abstract

ABSTRACT. Assume that f is a function defined on some interval I ⊂ R 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.

Inner variation and the SLi-functions

“…In particular, it was successfully used to give an alternative approach to the Henstock-Kurzweil integration in [5]. Moreover, while Lusin's condition (N ) was defined using the Lebesgue measure (see [8]), the strong Lusin condition was used to defined the SL-integral. It was shown that the SL-integral is equivalent to the Henstock-Kurzweil integral [10].…”