Compact \({\mathcal G}\)-fuzzy topological spaces
Do convergence and compactness properties of L-fuzzy topological spaces require the complete distributivity of the underlying lattice L? The author examines this question when L is a complete Boolean algebra because, if the answer were positive, the theory would be expected to have no significant application to probability theory. In fact, the paper provides a negative answer to the question for the special case considered. The author develops a convergence theory which does not require complete distributivity by replacing the prime prefilters of \textit{R. Lowen} [Gen. Topol. Appl. 10, 147-160 (1979; Zbl 0409.54008)] with a new concept of ultrafilters. As a result he can define a notion of compactness (called probabilistic compactness in the paper) which has properties parallel to ordinary compactness. For example, it can be characterized by a combination of completeness and pre-compactness. He also shows that, for any ordinary compact space X, the space of almost- everywhere defined X-valued random variables is probabilistic compact in a natural way.
complete distributivity of the underlying lattice
scientific article; zbMATH DE number 3913221
scientific article