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Daneš theorem in complete random normed modules
Journal of Inequalities and Applications volume 2014, Article number: 317 (2014)
Abstract
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 {L}^{0}convex topology. When the base space of the randomnormed module is trivial, our result automatically degenerates to the classicalcase.
MSC: 58E30, 47H10, 46H25, 46A20.
1 Introduction
In 1972, Daneš [1] presented the Daneš theorem. With the classical Ekeland variationalprinciple, Brøndsted [2] gave it a new proof in 1974.
Recently, Prof. Guo Tiexin and I [3] established the Ekeland variational principle for an {\overline{L}}^{0}valued function on a complete random metric space, where{\overline{L}}^{0} is the set of equivalence classes of extended realvaluedrandom variables on a probability space (\mathrm{\Omega},\mathcal{F},P). Based on this result, this paper establishes theDaneš theorem in a complete random normed module under the locally{L}^{0}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 (\mathrm{\Omega},\mathcal{F},P) is trivial, namely \mathcal{F}=\{\mathrm{\varnothing},\mathrm{\Omega}\}, 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.
2 Preliminary
Throughout this paper, (\mathrm{\Omega},\mathcal{F},P) denotes a probability space, K the real numberfield R or the complex number field C, N the set of positiveintegers, {\overline{L}}^{0}(\mathcal{F}) the set of equivalence classes of extended realvaluedrandom variables on Ω and {L}^{0}(\mathcal{F},K) the algebra of equivalence classes of Kvaluedℱmeasurable random variables on Ω under the ordinary scalar multiplication,addition and multiplication operations on equivalence classes, denoted by{L}^{0}(\mathcal{F}) when K=R.
Specially, {L}_{+}^{0}(\mathcal{F})=\{\xi \in {L}^{0}(\mathcal{F})\mid \xi \u2a7e0\}, {L}_{++}^{0}(\mathcal{F})=\{\xi \in {L}^{0}(\mathcal{F})\mid \xi >0\text{on}\mathrm{\Omega}\}.
As usual, \xi >\eta means \xi \u2a7e\eta and \xi \ne \eta, whereas \xi >\eta on A means {\xi}^{0}(\omega )>{\eta}^{0}(\omega ) a.s. on A for any A\in \mathcal{F} and ξ and η in{\overline{L}}^{0}(\mathcal{F}), where {\xi}^{0} and {\eta}^{0} are arbitrarily chosen representatives of ξand η, respectively.
For any A\in \mathcal{F}, {A}^{c} denotes the complement of A,\tilde{A}=\{B\in \mathcal{F}\mid P(A\mathrm{\Delta}B)=0\} denotes the equivalence class of A, where Δis the symmetric difference operation, {I}_{A} the characteristic function of A, and{\tilde{I}}_{A} is used to denote the equivalence class of{I}_{A}; given two ξ and η in{\overline{L}}^{0}(\mathcal{F}), and A=\{\omega \in \mathrm{\Omega}:{\xi}^{0}\ne {\eta}^{0}\}, where {\xi}^{0} and {\eta}^{0} are arbitrarily chosen representatives of ξand η, respectively, then we always write [\xi \ne \eta ] for the equivalence class of A and{I}_{[\xi \ne \eta ]} for {\tilde{I}}_{A}, one can also understand the implication of such notationsas {I}_{[\xi \le \eta ]}, {I}_{[\xi <\eta ]}, and {I}_{[\xi =\eta ]}.
For an arbitrary chosen representative {\xi}^{0} of \xi \in {L}^{0}(\mathcal{F},K), define the two ℱmeasurable random variables{({\xi}^{0})}^{1} and {\xi}^{0} by {({\xi}^{0})}^{1}(\omega )=\frac{1}{{\xi}^{0}(\omega )} if {\xi}^{0}(\omega )\ne 0, and {({\xi}^{0})}^{1}(\omega )=0 otherwise, and by {\xi}^{0}(\omega )={\xi}^{0}(\omega ), \mathrm{\forall}\omega \in \mathrm{\Omega}. Then the equivalence class {\xi}^{1} of {({\xi}^{0})}^{1} is called the generalized inverse of ξ andthe equivalence class \xi  of {\xi}^{0} is called the absolute value of ξ. It isclear that \xi \cdot {\xi}^{1}={I}_{[\xi \ne 0]}.
Definition 2.1 ([15])
An ordered pair (E,\parallel \cdot \parallel ) is called a random normed space (briefly, an RNspace) over K with base (\mathrm{\Omega},\mathcal{F},P) if E is a linear space and\parallel \cdot \parallel is a mapping from E to {L}_{+}^{0}(\mathcal{F}) such that the following three axioms are satisfied:

(1)
\parallel x\parallel =0 if and only if x=\theta (the null vector of E);

(2)
\parallel \alpha x\parallel =\alpha \parallel x\parallel, \mathrm{\forall}\alpha \in K and x\in E;

(3)
\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel, \mathrm{\forall}x,y\in E,
where the mapping \parallel \cdot \parallel is called the random norm on E and\parallel x\parallel is called the random norm of a vectorx\in E.
In addition, if E is left module over the algebra {L}^{0}(\mathcal{F},K) such that the following is also satisfied:

(4)
\parallel \xi x\parallel =\xi \parallel x\parallel, \mathrm{\forall}\xi \in {L}^{0}(\mathcal{F},K) and x\in E,
then such an RN space is called an RN module over K withbase (\mathrm{\Omega},\mathcal{F},P) and such a random norm \parallel \cdot \parallel is called an {L}^{0}norm on E.
Definition 2.2 ([16])
Let (E,\parallel \cdot \parallel ) be an RN module over K with base(\mathrm{\Omega},\mathcal{F},P). For any \epsilon \in {L}_{++}^{0}, let B(\epsilon )=\{x\in E\mid \parallel x\parallel \u2a7d\epsilon \} and {\mathcal{U}}_{\theta}=\{B(\epsilon )\mid \epsilon \in {L}_{++}^{0}\}. A set G\subset E is called {\mathcal{T}}_{c}open if for every x\in G there exists some B(\epsilon )\in {\mathcal{U}}_{\theta} such that x+B(\epsilon )\subset G. Let {\mathcal{T}}_{c} be the family of {\mathcal{T}}_{c}open subsets, then {\mathcal{T}}_{c} is a Hausdorff topology on E, called the locally{L}^{0}convex topology, denoted by {\mathcal{T}}_{c}.
Let (E,\parallel \cdot \parallel ) be an RN module over K with base(\mathrm{\Omega},\mathcal{F},P), {p}_{A}={\tilde{I}}_{A}\cdot p is called the Astratification of p foreach given A\in \mathcal{F} and p in E. The socalled stratificationstructure of E means that E includes every stratification of anelement in E. Clearly, {p}_{A}=\theta when P(A)=0 and {p}_{A}=p when P(\mathrm{\Omega}\setminus A)=0, which are both called trivial stratifications ofp. Further, when (\mathrm{\Omega},\mathcal{F},P) is a trivial probability space every element in Ehas merely the two trivial stratifications since \mathcal{F}=\{\mathrm{\Omega},\mathrm{\varnothing}\}; when (\mathrm{\Omega},\mathcal{F},P) 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 ([3])
Let X be a Hausdorff space and f:X\to {\overline{L}}^{0}(\mathcal{F}), then:

(1)
dom(f):=\{x\in X\mid f(x)<+\mathrm{\infty}\text{on}\mathrm{\Omega}\} is called the effective domain of f.

(2)
f is proper if f(x)>\mathrm{\infty} on Ω for every x\in X and dom(f)\ne \mathrm{\varnothing}.

(3)
f is bounded from below (resp., bounded from above) if there exists \xi \in {L}^{0}(\mathcal{F}) such that f(x)\ge \xi (accordingly, f(x)\le \xi) for any x\in X.
In all the vectorvalued extensions of the Ekeland variational principle, it is of keyimportance to properly define the lower semicontinuity for a vectorvalued function [17–19]. Recently, we have found that a kind of lower semicontinuity for{\overline{L}}^{0}valued functions is very suitable for the study ofconditional risk measures.
Definition 2.4 ([16])
Let (E,\parallel \cdot \parallel ) be an RN module over R with base(\mathrm{\Omega},\mathcal{F},P). A function f:E\to {\overline{L}}^{0}(\mathcal{F}) is called {\mathcal{T}}_{c}lower semicontinuous if epi(f) is closed in (E,{\mathcal{T}}_{c})\times ({L}^{0}(\mathcal{F}),{\mathcal{T}}_{c}).
There is a kind of countable concatenation property, which is concerned with the{L}^{0}module E itself and is very important for thetheory of RN module. Let us recall it.
Definition 2.5 ([10])
Let E be a left module over the algebra {L}^{0}(\mathcal{F},K). A formal sum {\sum}_{n\in N}{\tilde{I}}_{{A}_{n}}{x}_{n} is called a countable concatenation of a sequence\{{x}_{n}\mid n\in N\} in E with respect to a countable partition\{{A}_{n}\mid n\in N\} of Ω to ℱ. Moreover, a countable concatenation{\sum}_{n\in N}{\tilde{I}}_{{A}_{n}}{x}_{n} is well defined or {\sum}_{n\in N}{\tilde{I}}_{{A}_{n}}{x}_{n}\in E if there is x\in E such that {\tilde{I}}_{{A}_{n}}x={\tilde{I}}_{{A}_{n}}{x}_{n}, \mathrm{\forall}n\in N. A subset G of E is said to have thecountable concatenation property if every countable concatenation{\sum}_{n\in N}{\tilde{I}}_{{A}_{n}}{x}_{n} with {x}_{n}\in G for each n\in N still belongs to G, namely{\sum}_{n\in N}{\tilde{I}}_{{A}_{n}}{x}_{n} is well defined and there exists x\in G such that x={\sum}_{n\in N}{\tilde{I}}_{{A}_{n}}{x}_{n}.
Definition 2.6 ([16])
Let E be a left module over the algebra {L}^{0}(\mathcal{F}) and f a function from E to{\overline{L}}^{0}(\mathcal{F}), then:

(1)
f is {L}^{0}(\mathcal{F})convex if f(\xi x+(1\xi )y)\le \xi f(x)+(1\xi )f(y) for all x and y in E and \xi \in {L}_{+}^{0}(\mathcal{F}) such that 0\le \xi \le 1 (here we make the convention that 0\cdot (\pm \mathrm{\infty})=0 and \mathrm{\infty}\mathrm{\infty}=\mathrm{\infty}!).

(2)
f is said to have the local property if {\tilde{I}}_{A}f(x)={\tilde{I}}_{A}f({\tilde{I}}_{A}x) for all x\in E and A\in \mathcal{F}.
It is well known from [16] that f:E\to {\overline{L}}^{0}(\mathcal{F}) is {L}^{0}(\mathcal{F})convex iff f has the local property andepi(f) is {L}^{0}(\mathcal{F})convex.
3 Main results and proofs
Definition 3.1 Let (E,\parallel \cdot \parallel ) be an RN module over R with base(\mathrm{\Omega},\mathcal{F},P), z\in E and r\in {L}_{++}^{0}(\mathcal{F}). Denote the {\mathcal{T}}_{c}closed ball by
Definition 3.2 ([14])
Let (E,\parallel \cdot \parallel ) be an RN module over R with base(\mathrm{\Omega},\mathcal{F},P), {B}_{z}(r) a {\mathcal{T}}_{c}closed ball in E, y\in E\setminus {B}_{z}(r). Define the {L}^{0}(\mathcal{F})convex hull of \{y\}\cup {B}_{z}(r) by
Definition 3.3 Let (E,\parallel \cdot \parallel ) be an RN module over R with base(\mathrm{\Omega},\mathcal{F},P), ≤_{1} and ≤_{2} be bothorderings on E. Then ≤_{2} is finer than ≤_{1}, if
In [3], we established the precise form of the Ekeland variational principle on a{\mathcal{T}}_{c}complete RNmodule. Here we only need its generalform as follows.
Lemma 3.4 ([3])
Let(F,\parallel \cdot \parallel )be a{\mathcal{T}}_{c}complete RN moduleover R with base(\mathrm{\Omega},\mathcal{F},P)such that F has the countableconcatenation property, \phi :F\to {\overline{L}}^{0}(\mathcal{F})have the local property.IfG\subset Fis a{\mathcal{T}}_{c}closed subset with the countable concatenationproperty and\phi {}_{G}is proper, {\mathcal{T}}_{c}lower semicontinuous, and bounded from belowon G, then for each point{x}_{0}\in dom(\phi {}_{G}), there existsz\in Gsuch that the following are satisfied:

(1)
\phi (z)\le \phi ({x}_{0})\parallel z{x}_{0}\parallel;

(2)
for eachx\in Gsuch thatx\ne z, \phi (x)\nleqq \phi (z)\parallel xz\parallel, which means that z is a maximal element in(G,{\le}_{\phi}).
Remark 3.5 ([3])
The ordering {\le}_{\phi} on F is defined as follows:x{\le}_{\phi}y if and only if either x=y, or x and y\in dom(\phi ) are such that \parallel xy\parallel \le \phi (x)\phi (y).
Theorem 3.6 Let(X,\parallel \cdot \parallel )be a{\mathcal{T}}_{c}complete RN moduleover R with base(\mathrm{\Omega},\mathcal{F},P)such that X has the countableconcatenation property, F\subset Xa{\mathcal{T}}_{c}closed subset with the countable concatenationproperty, andz\in X\setminus F. Letr,R,\rho \in {L}_{++}^{0}(\mathcal{F})with0<r<R<\rhoon Ω, then thereexists{x}_{0}\in {\partial}_{c}(F)such that
and
whereR:=\bigwedge \{\parallel za\parallel :a\in F\}, and{\partial}_{c}(F)denotes the{\mathcal{T}}_{c}boundary of F.
Proof We can, without loss of generality, suppose z=0.
Let E:=F\cap {B}_{0}(\rho ).
Define an ordering \tilde{\le} on E as follows: {x}_{1}\phantom{\rule{0.2em}{0ex}}\tilde{\le}\phantom{\rule{0.2em}{0ex}}{x}_{2} if and only if {x}_{2}\in D(0,r,{x}_{1}). It is easy to check that \tilde{\le} is a partial ordering.
Define a function \phi :E\to {L}_{+}^{0}(\mathcal{F}) by \phi (x)=(\rho +r){(Rr)}^{1}\parallel x\parallel, \mathrm{\forall}x\in E.
Since F and {B}_{0}(\rho ) are {\mathcal{T}}_{c}closed and have the countable concatenation property, itfollows that E is {\mathcal{T}}_{c}closed and has the countable concatenation property.
For each A\in \mathcal{F}, one can have
which implies that φ has the local property.
Since φ is {\mathcal{T}}_{c}continuous, it is easy to see that φ is{\mathcal{T}}_{c}lower semicontinuous.
Then from Lemma 3.4, there exists a maximal element {x}_{0} in (E,{\le}_{\phi}).
We now prove that \tilde{\le} is finer than the ordering {\le}_{\phi}.
Let {x}_{1}, {x}_{2} be points in E such that {x}_{1}\phantom{\rule{0.2em}{0ex}}\tilde{\le}\phantom{\rule{0.2em}{0ex}}{x}_{2}. Then one can have {x}_{2}\in D(0,r,{x}_{1}); thus we can suppose
where t\in {L}_{+}^{0}(\mathcal{F}), 0\le t\le 1, and v\in {B}_{0}(r).
From (1), it follows that \parallel {x}_{2}\parallel \le (1t)\parallel {x}_{1}\parallel +t\parallel v\parallel, which implies
From Rr\le \parallel {x}_{1}\parallel \parallel v\parallel, one can have
Thus by (2) we have
which implies {x}_{1}\phantom{\rule{0.2em}{0ex}}{\le}_{\phi}\phantom{\rule{0.2em}{0ex}}{x}_{2}, and hence \tilde{\le} is finer than the ordering {\le}_{\phi}.
Since {x}_{0} is a maximal element in (E,{\le}_{\phi}) and \tilde{\le} is finer than {\le}_{\phi}, it is easy to check that {x}_{0} is a maximal element in (E,\tilde{\le}). Thus we have \parallel {x}_{0}\parallel \le \rho and \{{x}_{0}\}=D(0,r,{x}_{0})\cap E, which implies D(0,r,{x}_{0})\cap F=\{{x}_{0}\}.
For each x\in D(0,r,{x}_{0}), we can suppose x=t{x}_{0}+(1t)v, where v\in {B}_{0}(r). Thus we have
on Ω.
It is easy to see that for any y\in F\setminus E, \parallel y\parallel \le \rho does not hold. Hence we have {x}_{0}\in {\partial}_{c}(F). □
Remark 3.7 When the base space (\mathrm{\Omega},\mathcal{F},P) of the RN module is trivial, namely\mathcal{F}=\{\mathrm{\varnothing},\mathrm{\Omega}\}, 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).
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Yang, Y. Daneš theorem in complete random normed modules. J Inequal Appl 2014, 317 (2014). https://doi.org/10.1186/1029242X2014317
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DOI: https://doi.org/10.1186/1029242X2014317
Keywords
 random metric space
 random normed module
 locally {L}^{0}convex topology
 Ekeland variational principle