Abstract: Throughout this paper, （X,·） is assumed to be a real separable Banach space with a dual and separable X^*, we let (Ω，B,P) be a complete probability space, further, {B_n, n≥1} is an increase sub σfields filtration of B, and B=∨ B _n. First of all, we discussed some properties of support function, we used the support function and Riesz decomposition theorem and convergence theorem of real valued martingale (supermartingale, submartingale) to prove its Riesz decomposition theorem and convergence theorem, on the basis of which a sufficien t condition of Riesz decomposition of set-valued was given.

Key words: set-valued submartingale, weak convergence, Riesz decomposition