A generalized Henstock-Stieltjes integral involving division functions

We can consider the Riemann-Stieltjes integral $$ \int\limits_a^b f $$ dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral $$ \int\limits_a^b f $$ dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.