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Daneš theorem in complete random normed modules
Journal of Inequalities and Applications volume 2014, Article number: 317 (2014)
Based on the recent work of random metric theory, namely the Ekeland variationalprinciple on a complete random metric space, this paper studies the Daneštheorem in a complete random normed module. In this paper, we first present thenotion of finer ordering on a random normed module. Then we establish the Daneštheorem in a complete random normed module under the locally -convex topology. When the base space of the randomnormed module is trivial, our result automatically degenerates to the classicalcase.
MSC: 58E30, 47H10, 46H25, 46A20.
Recently, Prof. Guo Tiexin and I  established the Ekeland variational principle for an -valued function on a complete random metric space, where is the set of equivalence classes of extended real-valuedrandom variables on a probability space . Based on this result, this paper establishes theDaneš theorem in a complete random normed module under the locally-convex topology.
A random normed module is a random generalization of an ordinary normed space. Differentfrom ordinary normed spaces, random normed modules possess the rich stratificationstructure, which is introduced in this paper. It is this kind of rich stratificationstructure that makes the theory of random normed modules deeply developed and alsorenders it the most useful part of random metric theory [4–14]. When the probability is trivial, namely , our result reduces to the classical Daneš theorem.So our result is a nontrivial random extension.
The remainder of this article is organized as follows: in Section 2 we give somenecessary definitions and in Section 3 we give the main results and proofs.
Throughout this paper, denotes a probability space, K the real numberfield R or the complex number field C, N the set of positiveintegers, the set of equivalence classes of extended real-valuedrandom variables on Ω and the algebra of equivalence classes of K-valuedℱ-measurable random variables on Ω under the ordinary scalar multiplication,addition and multiplication operations on equivalence classes, denoted by when .
Specially, , .
As usual, means and , whereas on A means a.s. on A for any and ξ and η in, where and are arbitrarily chosen representatives of ξand η, respectively.
For any , denotes the complement of A, denotes the equivalence class of A, where Δis the symmetric difference operation, the characteristic function of A, and is used to denote the equivalence class of; given two ξ and η in, and , where and are arbitrarily chosen representatives of ξand η, respectively, then we always write for the equivalence class of A and for , one can also understand the implication of such notationsas , , and .
For an arbitrary chosen representative of , define the two ℱ-measurable random variables and by if , and otherwise, and by , . Then the equivalence class of is called the generalized inverse of ξ andthe equivalence class of is called the absolute value of ξ. It isclear that .
Definition 2.1 ()
An ordered pair is called a random normed space (briefly, an RNspace) over K with base if E is a linear space and is a mapping from E to such that the following three axioms are satisfied:
if and only if (the null vector of E);
, and ;
where the mapping is called the random norm on E and is called the random norm of a vector.
In addition, if E is left module over the algebra such that the following is also satisfied:
, and ,
then such an RN space is called an RN module over K withbase and such a random norm is called an -norm on E.
Definition 2.2 ()
Let be an RN module over K with base. For any , let and . A set is called -open if for every there exists some such that . Let be the family of -open subsets, then is a Hausdorff topology on E, called the locally-convex topology, denoted by .
Let be an RN module over K with base, is called the A-stratification of p foreach given and p in E. The so-called stratificationstructure of E means that E includes every stratification of anelement in E. Clearly, when and when , which are both called trivial stratifications ofp. Further, when is a trivial probability space every element in Ehas merely the two trivial stratifications since ; when is arbitrary, every element in E can possessarbitrarily many nontrivial intermediate stratifications. It is this kind of richstratification structure of RN modules that makes the theory of RNmodules deeply developed and also renders it the most useful part of random metrictheory.
To introduce the main results of this paper, let us first recall the following.
Definition 2.3 ()
Let X be a Hausdorff space and , then:
is called the effective domain of f.
f is proper if on Ω for every and .
f is bounded from below (resp., bounded from above) if there exists such that (accordingly, ) for any .
In all the vector-valued extensions of the Ekeland variational principle, it is of keyimportance to properly define the lower semicontinuity for a vector-valued function [17–19]. Recently, we have found that a kind of lower semicontinuity for-valued functions is very suitable for the study ofconditional risk measures.
Definition 2.4 ()
Let be an RN module over R with base. A function is called -lower semicontinuous if is closed in .
There is a kind of countable concatenation property, which is concerned with the-module E itself and is very important for thetheory of RN module. Let us recall it.
Definition 2.5 ()
Let E be a left module over the algebra . A formal sum is called a countable concatenation of a sequence in E with respect to a countable partition of Ω to ℱ. Moreover, a countable concatenation is well defined or if there is such that , . A subset G of E is said to have thecountable concatenation property if every countable concatenation with for each still belongs to G, namely is well defined and there exists such that .
Definition 2.6 ()
Let E be a left module over the algebra and f a function from E to, then:
f is -convex if for all x and y in E and such that (here we make the convention that and ).
f is said to have the local property if for all and .
It is well known from  that is -convex iff f has the local property and is -convex.
3 Main results and proofs
Definition 3.1 Let be an RN module over R with base, and . Denote the -closed ball by
Definition 3.2 ()
Let be an RN module over R with base, a -closed ball in E, . Define the -convex hull of by
Definition 3.3 Let be an RN module over R with base, ≤1 and ≤2 be bothorderings on E. Then ≤2 is finer than ≤1, if
In , we established the precise form of the Ekeland variational principle on a-complete RN-module. Here we only need its generalform as follows.
Lemma 3.4 ()
Letbe a-complete RN moduleover R with basesuch that F has the countableconcatenation property, have the local property.Ifis a-closed subset with the countable concatenationproperty andis proper, -lower semicontinuous, and bounded from belowon G, then for each point, there existssuch that the following are satisfied:
for eachsuch that, , which means that z is a maximal element in.
Remark 3.5 ()
The ordering on F is defined as follows: if and only if either , or x and are such that .
Theorem 3.6 Letbe a-complete RN moduleover R with basesuch that X has the countableconcatenation property, a-closed subset with the countable concatenationproperty, and. Letwithon Ω, then thereexistssuch that
where, anddenotes the-boundary of F.
Proof We can, without loss of generality, suppose .
Define an ordering on E as follows: if and only if . It is easy to check that is a partial ordering.
Define a function by , .
Since F and are -closed and have the countable concatenation property, itfollows that E is -closed and has the countable concatenation property.
For each , one can have
which implies that φ has the local property.
Since φ is -continuous, it is easy to see that φ is-lower semicontinuous.
Then from Lemma 3.4, there exists a maximal element in .
We now prove that is finer than the ordering .
Let , be points in E such that . Then one can have ; thus we can suppose
where , , and .
From (1), it follows that , which implies
From , one can have
Thus by (2) we have
which implies , and hence is finer than the ordering .
Since is a maximal element in and is finer than , it is easy to check that is a maximal element in . Thus we have and , which implies .
For each , we can suppose , where . Thus we have
It is easy to see that for any , does not hold. Hence we have . □
Remark 3.7 When the base space of the RN module is trivial, namely, our result automatically degenerates to the classicalDaneš theorem. So our result is a nontrivial random extension.
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This work was supported by the ‘New Start’ Academic Research Projects ofBeijing Union University (No. Zk10201304).
The author declares to have no competing interests.
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Cite this article
Yang, Y. Daneš theorem in complete random normed modules. J Inequal Appl 2014, 317 (2014). https://doi.org/10.1186/1029-242X-2014-317
- random metric space
- random normed module
- locally -convex topology
- Ekeland variational principle