Variational measures and the Kurzweil-Henstock integral

Rok, strany: 2009, 731 - 752

For a given continuous function $F$ on a compact interval $E$ in the set $\mathbb{R}$ of reals the problem is how to describe the ``total change" of $F$ on a set $M\subset E$. 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 $f$.

ISO 690:

Schwabik, Š. 2009. Variational measures and the Kurzweil-Henstock integral. In Mathematica Slovaca, vol. 59, no.6, pp. 731-752. 0139-9918. DOI: https://doi.org/10.2478/s12175-009-0160-1

APA:

Schwabik, Š. (2009). Variational measures and the Kurzweil-Henstock integral. Mathematica Slovaca, 59(6), 731-752. 0139-9918. DOI: https://doi.org/10.2478/s12175-009-0160-1