On Variational Measures Related to Some Bases

Abstract

We extend, to a certain class of differentiation bases, some results on the variational measure and the δ-variation obtained earlier for the full interval basis. In particular the theorem stating that the variational measure generated by an interval function is σ-finite whenever it is absolutely continuous with respect to the Lebesgue measure is extended to any Busemann-Feller basis.

On the uniform strong Lusin condition

“…An analogous result for ρ-regular Kurzweil-Henstock integration had been proved earlier by Di Piazza [8] and Skvortsov [18]. Bongiorno, Di Piazza and Skvortsov extended their Theorem 3 to the case of one-dimensional integration with respect to some derivation bases with the so-called BusemannFeller property [6]. Among the bases included were all P-adic bases with the…”

Absolutely continuous variational measures of Mawhin’s type

“…A nonnegative real-valued function δ(·) is called a gage if its null set x : δ(x) = 0 ∈ K . 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]).…”

“…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]).…”

“…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]). 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,…”

“…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.…”