Sufficient conditions for convergence of riemann sums for function space defined by the $k$-modulus of continuity

Abstract

For classes of functions defined by $k$-th moduli of continuity, calculated in the norm of the symmetric space $X$, an estimate of the difference of two members of the sequence of operators of Riemann sums on a set of large measure is given. The estimate is given in terms of a fundamental function of the space $X$. Using this result, for a function $f$ from a given symmetric space, sufficient conditions are given for the almost everywhere convergence of the sequence of operators of Riemann sums to the Lebesgue integral of a given function.