Abstract:

Some certain examples were mentioned in order to prove that  not all the setvalued submartingle could be decomposed even in one dimension real space. In other words, the setvalued submartingle means the sum of setvalued martingle and setvalued submartingle. A new definition about the Riesz decomposition of setvalued submartingle was presented. We  proved that the set valued submartingle possesses the new form of definition in one dimension real space. In order to illustrate that the new definition is not available in two\|dimension real space, some examples were given here. At last, we gave and proved that the sufficient and necessary conditions of this new Riesz decomposition.

Key words: setvalued (sub) martingale, KuratowskiMosco convergence, Riesz decomposition； support function