Several properties of generalized multivariate integrals and theorems of the du Bois-Reymond type for Haar series

Several properties of generalized multivariate integrals are considered. In the two-dimensional case the consistency of the regular Perron integral is proved, as well as the consistency of a generalized integral solving the problem of the recovery of the coefficients of double Haar series in a certain class. Several generalizations of Skvortsov's well-known theorem are obtained as consequences, for instance, the following result: if a double Haar series converges for some $\rho\in(0,1/2]$ $\rho$-regularly everywhere in the unit square to a finite function that is Perron-integrable in the $\rho$-regular sense, then the series in question is the Fourier–Perron series of its sum.
Bibliography: 20 titles.