The Role of Harnack Extension in the Kurzweil-Stieltjes Sense: Integrating Functions over arbitrary Subsets

Umi Mahnuna Hanung

Published 2022-07-10Version 1

The Harnack extension principle discussing a sufficient condition for the integrable functions on particular subsets of $(a,b)$ to be integrable on $[a,b]$ is already included in the Kurzweil-Henstock integral (see e.g. Theorem 1.1). The Kurzweil-Stieltjes integral reduces to the Kurzweil-Henstock integral whenever the integrator is an identity function. It is known that if the integrator $F$ is discontinuous on $[c,d]\subset[a,b]$, then the values of the Kurzweil-Stieltjes integrals $$\int_c^d[dF]g,\ \int_{[c,d]}[dF]g,\ \int_{[c,d)}[dF]g,\ \int_{(c,d]}[dF]g,\ {\rm and}\ \int_{(c,d)}[dF]g$$ need not coincide (see [37, Section 5]). This indicates that the Harnack extension principle in the Kurzweil-Henstock integral cannot be valid any longer for the Kurzweil type Stieltjes integrals with discontinuous integrators. The concepts of equiintegrability and equiregulatedness are pivotal to the notion of Harnack extension for the Kurzweil-Stieltjes integration. Moreover, it is also known that, in general, the existence of the integral $\int_a^b[dF]g$ does not (even in the case of the identity integrator $F(x)=x$) always imply the existence of the integral $\int_{T}[dF]g$ for every subset $T$ of $[a,b]$. This follows from the well-known fact that, if e.g. $T\subset[a,b]$ is not measurable, then the existence of the Lebesgue integral $\int_a^b g [dt]$ (which is a particular case of the Kurzweil-Henstock one) does not imply, in general, that also the integral $\int_T g [dt]$ exists. Therefore, besides having an interest in constructing Harnack extension principle for the abstract Kurzweil-Stieltjes integral, the aim of this paper is to investigate the existence of the integrals $\int_{T}[dF]g$ for arbitrary closed subsets $T$ of an elementary set $E$.