The Riesz Decomposition of Set-Valued Superpramart

The paper proves the convergence theorem of set-valued Superpramart in the sense of weak convergence under the X* separable condition. Using support function and results about real-valued Superpramart, we give the Riesz decomposition of set-valued Superpramart.


Introduction
Reference [1] gives Riesz decomposition of set-valued supermartingale in real space and promotes the results to reflexive Banach spaces (reference [2]).Reference [2] gives the counter-example that not all of the set-valued martingale has Riesz decomposition in a two-dimensional plane case.The fundamental reason is the defects of algebraic operation on hyperspace.Therefore, the research can pursue the unstrict sense of Riesz decomposition instead of studying various sense of Riesz decomposition.Reference [2] shows the other Riesz decomposition of set-valued supermartingale in real space.Reference [3] gives Riesz decomposition of set-valued supermartingale in the general Banach space under the X * separable condition.References [4] and [5] research Riesz decomposition of set-valued submartingale in the general Banach space.Reference [6] studies Riesz decomposition in weak set-valued Amart.Reference [7]- [9] gives every sense of Riesz decomposition of set-valued Pramart in the general Banach space under the X * separable condition.Reference [10] studies the problems of Riesz decomposition of set-valued Pramart.All of the above studies have given the necessary and sufficient conditions for Riesz decomposition.The research of every sense of Riesz decomposition of set-valued Superpramart is still rare.
The paper firstly demonstrates convergence theorem that set-valued Superpramart is in the sense of weak convergence under the X * separable condition.On this basis, using support function and results about realvalued Superpramart, we give a class of Riesz decomposition of set-valued Superpramart.

Method
Assume (X,‖ •‖ ) as a separable Banach space, D 1 is X-fan subset of the columns that can be condensed.X * is the dual space X.X * is separable.
is X * -fan subset of the columns that can be condensed, remember Assume (Ω,,P) is a complete probability space.{ n , n ≥ 1} is the 's rise σ , and  = ∨ n , ( ) represents that the value of ( ) fc P X is all the integrable bounded random set.
In order to write simply, often eliminating the almost certainly established under the meaning of the equations, inequalities and tag contain relations sense "a.s.
The uniqueness is proved by the following: Supposing (1) , 1

S. Y. Li et al.
From Lemma 1, Lemma 6 and reference [2] , by the separability of X * and the continuity of X * support function, we know from reference [2] corollary 1.4.1 that: ( | ) , and sup

Conclusion
The paper proves the convergence theorem of Superpramart in the sense of weak convergence.And on the basis of this certificate, through the support function and the results of real-valued Superpramart, we give the one of Riesz decomposition forms of set-valued Superpramart.It provides new ideas for the research of Riesz decomposition.
and from the list of D * , we know the little-known set N 1 real-valued consistent Subpramart, and from reference [2] lemma 4.4.2,we know the little-known set N 2 exists, lemma 4.1.3,it's easy to know the above equation is the Riesz decomposition of real-valued Superpramart *