On a condition of existence of non-anticipating selections

摘要

The hereditary selections of multi-functions play an important role in the theory of diferential games in connection with the construction of resolving quasi-strategies. The existence of a non-anticipating selection of a non-anticipating multi-function is considered. In most cases important for applications, it is known that any non-anticipating multi-function with non-empty compact values has a non-anticipating selection. Namely, the result is valid when the non-anticipation property is defined by a totally ordered family in the domain of ”time” variable. In this note, we show that the condition is essential: when the family is not totally ordered, there exists a hereditary multi-function with non-empty compact values that has no non-anticipating selections.