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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.

Reference [^{*} separable condition. References [^{*} separable condition. Reference [

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 real- valued Superpramart, we give a class of Riesz decomposition of set-valued Superpramart.

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. D^{*} = ^{*}-fan subset of the columns that can be condensed, remember

Any

_{n} weak convergence in A, denote as

If

Assume (W,G,P) is a complete probability space. {G_{n}, n ≥ 1} is the G’s rise_{n}, _{n}, G_{n}, n ≥1}

as adapted random set columns. If_{n} can be measured by G_{n}. If

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.”, {F_{n}, G_{n}, n ≥ 1}. {x_{n}, G_{n}, n ≥ 1} is often referred as_{n}, n ≥ 1}.

Definition 1 Supposing

1) If

2) If

Definition 2 Supposing

If

Definition 3 Supposing

Definition 4 We call set-valued Superpramart

Lemma 1 [

1)

2)

Lemma 2 Supposing

Proof:

Lemma 3 [

1)

2)

Lemma 4 Supposing

dom set

Proof:

and because

ence [^{*}, we know

Lemma 5 Supposing

torization

Proof: From Lemma 2, we know

and

The uniqueness is proved by the following: Supposing

Lemma 6 Supposing

Proof: From Lemma 4, we know the random set

noting

consistent Subpramart, then from reference [

Lemma 7 Supposing

1)

2)_{n} and E(F|G_{n})(n ≥ 1) are homothetic, where

Proof: We prove

1)

Then, S(x^{*},F_{n}) = S(x^{*},G_{n}) + S(x^{*},Z_{n}).

From Lemma 1, Lemma 6 and reference [^{*} and the continuity of X^{*} support function, we know from reference [_{n} and E(F|G_{n}) are homothetic.

2)_{n}), it’s easy to know {G_{n}, n ≥ 1} is the value martingale of

pramart of

S(x^{*},F_{n}) = S(x^{*},G_{n}) + S(x^{*},Z_{n})

S(x^{*},Z_{n}) = S(x^{*},F_{n}) – S(x^{*},G_{n})

It’s easy to prove

E(F_{t}|G_{s}) = G_{s} + E(Z_{t}|G_{s}), sÎT, tÎT (s), from reference [

Then, we know

S(x_{i}^{*},Z_{n}) = S(x_{i}^{*},F_{n}) – S(x_{i}^{*},G_{n}), x^{*}_{i}ÎD^{*}

=S(x_{i}^{*},F_{n}) – S(x_{i}^{*},E(F|G_{n}))

=S(x_{i}^{*},F_{n}) – E(S (x_{i}^{*},F)|G_{n}), and from the list of D^{*}, we know the little-known set N_{1}, and

Noting

consistent Subpramart, and from reference [_{2} exists,

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.

Shuyuan Li,Gaoming Li,Hang Dong,Caoshan Wang, (2016) The Riesz Decomposition of Set-Valued Superpramart. Journal of Applied Mathematics and Physics,04,1275-1279. doi: 10.4236/jamp.2016.47134