On the value of a stopped set function process

For certain types of stochastic processes {X_n | n ∈ $\text{N}$}, which are integrable and adapted to a nondecreasing sequence of σ-algebras $\text{F}$_n on a probability space (Ω, $\text{F}$, P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {$\text{F}$_n | n ∈ $\text{N}$}, when is$\text{sup}{}_{\mathrm{s}}\text{∫}{}_{\mathrm{\Omega }}\text{| X}{}_{\mathrm{\sigma }}\text{| dP}$finite, and when is there a stopping time $\text{σ}$ 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.