A Vitali-type theorem for vector lattice-valued modulars with respect to filter convergence is proved. Some applications are given to modular convergence theorems for moment operators in the vector lattice setting, and also for the Brownian motion and related stochastic processes.
Abstract:We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.
MSC:40A35, 41A35, 46E30
ABSTRACT. Some Schur, Nikodým, Brooks-Jewett and Vitali-Hahn-Saks-type theorems for ( )-group-valued measures are proved in the setting of filter convergence. Finally we pose an open problem.
ABSTRACT. Some aspects of the theory of order and (D)-convergence in ( )-groups with respect to ideals are investigated. Moreover some new Basic Matrix Theorems are proved.
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