On the value of a stopped set function process

For certain types of stochastic processes {Xn n [set membership, variant] }, which are integrable and adapted to a nondecreasing sequence of [sigma]-algebras n on a probability space ([Omega], , P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {n n [set membership, variant] }, when issups [integral operator][Omega] X[sigma] dP finite, and when is there a stopping time 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 [sigma]-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research.Martingale submartingale amart semiamart set function process stopping times