ON POSITIVE RANDOM OBJECTS

The idea of defining the expectation of a random variable as its integ ral with respect to a probability measure is extended to certain latti ce-valued random objects and basic results of integration theory ale g eneralized. Conditional expectation is defined and its properties are developed, Lattice-valued martingales are also studied and convergence of sub-and supermartingales and the Optional Sampling Theorem are pro ved. A martingale proof of the Strong Law of Large Numbers is given. A n extension of the lattice is also studied. Studies of some applicatio ns, such as on random compact convex sets in R-n and on random positiv e upper semicontinuous functions, are carried out, Where the generaliz ed integral is compared with the classical definition. The results are also extended to the case where the probability measure is replaced b y a sigma-finite measure.