Geodesic
On a~family of analogs of the Perron--Stieltjes integral
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2011), pp. 95-106

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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.
Keywords: regulated function, Riemann–Stieltjes integral, Perron–Stieltjes integral. 
@article{VUU_2011_3_a8,
     author = {V. I. Rodionov},
     title = {On a~family of analogs of the {Perron--Stieltjes} integral},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {95--106},
     publisher = {mathdoc},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2011_3_a8/}
}
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V. I. Rodionov. On a~family of analogs of the Perron--Stieltjes integral. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2011), pp. 95-106. http://geodesic.mathdoc.fr/item/VUU_2011_3_a8/