Set indexed strong martingales and path independent variation (Q1012109)

Let \((T,\tau)\) be a non-void sigma-compact connected topological space. The author studies strong martingales indexed by an indexed collection \({\mathfrak A}\). By definition, \({\mathfrak A}\) is a family of compact connected subsets of \(T\), satisfying certain conditions. (This notion was introduced by \textit{G. Ivanoff} and \textit{E. Merzbach} [Set-indexed martingales (1999; Zbl 0948.60039)].) It was shown by \textit{R. Cairoli} and \textit{J. B. Walsh} [Acta math. 134, 111--183 (1975; Zbl 0334.60026)] that two-parameter strong martingales satisfying certain conditions show path independent variation (p.i.v.). The author of the present paper extends their results to set-indexed strong martingales (satisfying certain conditions). In the last section the author investigates the p.i.v. property for set-indexed processes having independent increments.