On a family of analogs of the Perron–Stieltjes integral

We define the concept of a quasi-integral for two regulated functions defined on a segment and for a special parameter called a defect. In case there exists the Riemann–Stieltjes integral, there is a quasi-integral for any defect, and all quasi-integrals are equal. The Perron–Stieltjes integral, if it exists, coincides with one of quasi-integrals where the defect is defined in a special way. We give proofs of necessary and sufficient conditions for the existence of quasi-integrals and of their basic properties, in particular, of the analogue of the formula of integration by parts.