Summable and integrable functions with respect to any Loomis-system (Q1088018)

In this paper we generalize the process of prolongation of a Daniell functional, starting in this case with a vector lattice of real functions and a nonnegative linear functional defined on it. The initial elemental integral is extended to a class of summable functions by the previous introduction of the appropriate oscillation integrals, and then we consider the integrable functions, which form the widest possible class of those to which the functional may be prolonged.