On the value of a stopped set function process

AbstractFor certain types of stochastic processes {Xn | n ∈ N}, which are integrable and adapted to a nondecreasing sequence of σ-algebras Fn on a probability space (Ω, F, P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {Fn | n ∈ N}, when issups ∫Ω | Xσ | dP finite, and when is there a stopping time σ for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research