The Alpha-integral of Stieltjes type

The equivalence classes of regulated functions, which differ only on the points of discontinuity, are considered. On these classes the integral of Riemann–Stieltjes is extenced; one modification of Perron–Stieltjes integral (called alpha-integral) is introduced for them. The integration by parts theorem, the substitution theorem, the change of order of integration theorem, the convergence theorems for these integrals are proved.