McShane Integrability Using Variational Measure

If f : [a, b] → R is McShane integrable on [a, b], then f is McShane integrable on every Lebesgue measurable subset of [a, b]. However, integrability of a real-valued function on [a, b] does not imply McShane integrability on any E ⊆ [a, b]. In this paper, we give a characterization for the McShane integrability of f : [a, b] → R over E ⊆ [a, b] using concept of variational measure.

A version of Hake’s theorem for Kurzweil–Henstock integral in terms of variational measure

We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil–Henstock-type integral related to this basis. We prove a version of Hake’s theorem in terms of a variational measure.

Variational measures and the Kurzweil-Henstock integral

For a given continuous function F on a compact interval E in the set ℝ of reals the problem is how to describe the “total change” of F on a set M ⊂ E. Full variational measures W F(M) and V F(M) (see Section 2) in the sense presented by B. S. Thomson are introduced in this work to this aim. They are generated by two slightly different interval functions, namely the oscillation of F over an interval and the value of the additive interval function generated by F, respectively. They coincide with the concept of classical total variation if M is an interval and they are zero if on the set M the function F is of negligible variation.The Kurzweil-Henstock integration is shortly described and some of its properties are studied using the variational measure W F(M) for the indefinite integral F of an integrable function f.