  Various kinds of Stieltjes integrals using gauge integration have become
highly popular in the field of differential equations and other applications.
In the theories of integration and of ordinary differential equations,
convergence theorems provide one of the most widely used tools. The Harnack
extension principle, which discusses a sufficient condition for
Kurzweil-Henstock integrable functions on particular subsets of $(a,b)$ to be
integrable on $[a,b]$, is a key step to supply convergence theorems. The
Kurzweil-Stieltjes integral reduces to the Kurzweil-Henstock integral whenever
the integrator is an identity function. In general, 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. Hence,
the Harnack extension principle in the Kurzweil-Henstock integral cannot be
valid any longer for the Kurzweil-type Stieltjes integrals with discontinuous
integrators. The new concepts of equi-integrability and equiregulatedness are
pivotal to the notion of the Harnack extension principle for the
Kurzweil-Stieltjes integration.
  Moreover, the existence of the integral $\int_a^b[dF]g$ does not (even in the
case of the identity integrator) 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 $\int_a^b g [dt]$ does not imply that the
integral $\int_T g [dt]$ exists. Therefore, besides constructing the Harnack
extension principle for the abstract Kurzweil-Stieltjes integral, the aim of
this paper is also to demonstrate its role in guaranteeing the existence of the
integrals $\int_{T}[dF]g$ for arbitrary subsets $T$ of an elementary set $E$.
