A Skorokhod topology for a class of set-indexed functions (Q2726262)

The different topologies defined by \textit{A. V. Skorokhod} [Teor. Veroyatn. Primen. 1, 289-319 (1956; Zbl 0074.33802)] on the space of right-continuous functions with left-limits on the unit interval have been thoroughly studied. These topologies can be characterized using graphs of functions and the four topologies appear as the four possible combinations of two metrics (the Hausdorff metric and the parametric representation metric) applied to two kinds of graphs (complete and incomplete). However, for functions defined on spaces more general than the unit interval or the real line the situation becomes much more complicated and clearly the first problem is to find a suitable generalization of the space \(D[0,1]\) of real functions having left limits and right continuity. NEWLINENEWLINENEWLINEIn this paper different extensions are summarized: (1) \textit{M. L. Straf} [in: Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 2, 187-221 (1972; Zbl 0255.60019)] studied a general space \(D(T)\) of functions with possible jumps on an arbitrary space \(T\), and taking his cue from Skorokhod he presented an associated collection of metrics for \(D(T)\); (2) \textit{R. F. Bass} and \textit{R. Pyke} [Ann. Probab. 13, 860-884 (1985; Zbl 0585.60007)] dealt with a space \({\mathcal D}({\mathcal A}),\) where \({\mathcal A},\) the domain of definition of the functions, is a family of Borel subsets of the \(n\)-dimensional unit cube \([0,1]^{n}.\) The goal of this note is to unify the approaches by Straf, Bass and Pyke by showing that the resulting topologies on a class of outer continuous with inner limits functions are equivalent. The framework presented here is very general and can be applied to many situations.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].