Optional sampling of submartingales with scanned index sets (Q1187525)

It is known, that Doob's optional sampling theorem is not valid, when the submartingale is defined with respect to partially ordered index sets. The authors give an additional requirement on the structure of partially ordered sets, and partially ordered sets with this structure are called scanned sets. They show that scanned sets cover previously used structures like index sets with uncountable tree structure and sets which are dense from above. The authors prove Doob's theorem for partially ordered sets with scanning property. The authors show that the notion of a scanned set can be applied to sequential sampling plans.