Infinite Variation and Derivatives inRm

Abstract

We show that functions defined on Vitali families of sets in R m either have upper derivative finite almost everywhere or their variation is infinite on a large collection of null sets. As an example of applications, we show how to improve a characterisation of primitives of certain conditionally convergent integrals.

On the Minimal Solution of the Problem of Primitives

“…The idea of considering appropriate variational measures to characterize the primitives of some integral has been also used in [2,8,9] for many multidimensional integrals and in [5] for the Henstock-dyadic integral and for the Henstock-symmetric integral.…”

Absolutely continuous variational measures of Mawhin’s type

“…If we put K = {∅}, then we obtain the classical gage which is used in the theory of Henstock-type integrals (see, for example, [6,8,17,21,26,27,28,33,39,43,44]). If we put K to be the family of all thin sets, then we get the gage which is used in the study of F -and BV -integrals (see, for example, [3,10,12,13,14,15,16,29,30,31,32]). At last, if K is the family of all negligible sets, we obtain the notion of essential gage (see, for example, [3,5,10,13,32]).…”

“…If we put K to be the family of all thin sets, then we get the gage which is used in the study of F -and BV -integrals (see, for example, [3,10,12,13,14,15,16,29,30,31,32]). At last, if K is the family of all negligible sets, we obtain the notion of essential gage (see, for example, [3,5,10,13,32]). …”

“…It turned out that the absolute continuity and σ-finiteness play the central role in the theory of these measures. Absolutely continuous variational measures characterize primitives of conditionally convergent integrals (see, for example, [3,4,6,7,8,9,14,15,17,18,27,28,31,33,36,37,38,39,40,45,48,49]). At the same time, σ-finiteness of a variational measure gives some information about differentiability properties of the set function that determines this measure (see [2,4,6,7,10,13,21,24,32,35,36,40,41,42,44,49]).…”

“…At the same time, σ-finiteness of a variational measure gives some information about differentiability properties of the set function that determines this measure (see [2,4,6,7,10,13,21,24,32,35,36,40,41,42,44,49]). It is also well-known that, unlike general situation, the absolute continuity of a variational measure implies its σ-finiteness (see [3,4,6,7,9,12,14,15,17,19,27,31,33,36,39,40,44,46,47,49]). This relation between the absolute continuity and σ-finiteness motivated several authors to find characterizations of σ-finite variational measures.…”