On a certain series
We consider processes Xₜ with values in and “time” index t in a subset A of the unit cube. A natural condition of boundedness of increments is assumed. We give a full characterization of the domains A for which all such processes are a.e. continuous. We use the notion of Talagrand’s majorizing measure as well as geometrical Paszkiewicz-type characteristics of the set A. A majorizing measure is constructed.
The notion of -convergence of a sequence of functions is stronger than pointwise convergence and weaker than uniform convergence. It is inspired by the investigation of ill-posed problems done by A.N. Tichonov. We answer a question posed by M. Katětov around 1970 by showing that the only analytic metric spaces for which pointwise convergence of a sequence of continuous real valued functions to a (continuous) limit function on implies -convergence are -compact spaces. We show that the assumption...