Some Convergence Results for Partial Maps

Agata Caserta, Roberto Lucchetti

In this paper we analyze some aspects of a new notion of convergence for nets of partial maps, introduced in [8]. In particular, we show that the introduced bornological convergence reduces to a natural uniform convergence relative to the bornology when the partial maps have a common domain. We then provide a new notion of upper convergence, which looks much more manageable than the original one. We show that the two notions, though different in general cases, do agree for \emph{sequences} of strongly uniformly continuous (relative to the bornology) partial maps. More generally, coincidence for nets is shown in case the target space of the maps is totally bounded. This last result is interesting in view of possible applications, since partial maps are usually utility functions, thus when dealing with general models, monotone transformations valued in $[0,1]$ give rise to the same utility functions.