Asymptotic behavior of continuous set-indexed martingales

In this paper. Following the Knight's approach, we solve a convergence problem Tot set-indexed martingales. For this purpose, we first define a tightness criterion For set-indexed continuous processes. The core of this: characterization is connected with a weaker definition of continuity and hence the use of the corresponding topology, and with the Fact that indices take values in a semilattice of closed subsets. Then, we give an effective lightness criterion by means of an estimate For a majorizing measure defined on the space. We finally prove under this set-indexed framework a theorem similar to the Knight's.