Dominated Convergence and Egorov Theorems for Filter Convergence - Kadets, V.; Leonov, A. (Журнал математичної фізики, аналізу, геометрії, 2007)
We study the filters, such that for convergence with respect to this filtersthe Lebesgue dominated convergence theorem and the Egorov theorem onalmost uniform convergence are valid (the Lebesgue filters and the Egorovfilters, respectively). Some characterizations of the Egorov and the Lebesgue filters are given. It is shown that the class of Egorov filters is a proper subset of the class of Lebesgue filters, in particular, statistical convergence filter is the Lebesgue but not the Egorov filter. It is also shown that there are no free Lebesgue ultrafilters. Significant attention is paid to the filters generated by a matrix summability method. Mathematics Subject Classification: 28A20, 54A20, 40C05.