Leonov O. Generalized types of convergence in problems of Banach space theory and measure theory

Theorems about cluster points and subsequences for filters are established. Filters for which the Egorov theorem holds and filters (called Lebesgue) for which the dominated convergence theorem holds are characterized. It is proved that filters for which the dominated convergence theorem on [0,1] with Lebesgue measure holds and filters for which the Rainwater extremal test theorem is true form the same class, which is strictly bigger than the class of all Lebesgue filters. Schur l1 theorem for filter convergence is investigated. Series convergence is generalized on filters and a filter analogue of Riemann theorem is studied.