Daneš theorem in complete random normed modules

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 real-valuedrandom 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 real-valuedrandom variables on Ω and $L^0(\mathcal{F},K)$ 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$L^0(\mathcal{F})$ when $K=R$.

Specially, $L_{+}^0(\mathcal{F})=\{\xi \in L^0(\mathcal{F})\mid \xi ⩾0\}$, $L_{++}^0(\mathcal{F})=\{\xi \in L^0(\mathcal{F})\mid \xi >0\text{ on }\mathrm{\Omega }\}$.

As usual, $\xi >\eta$ means $\xi ⩾\eta$ 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 ℱ-measurable 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 vector$x\in E$.

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 $B(\epsilon )=\{x\in E\mid \parallel x\parallel ⩽\epsilon \}$ 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 A-stratification of p foreach given $A\in \mathcal{F}$ and p in E. The so-called 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 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${\overline{L}}^0$-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 and$epi(f)$ 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

$$B_z(r):=\{x\in E:\parallel x-z\parallel \le r\}.$$

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

$$D(z,r,y):=\{tb+(1-t)y:b\in B_z(r),t\in L_{+}^0(\mathcal{F})\text{ and }0\le t\le 1\}.$$

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

$$x{\le }_{2}y\Rightarrow x{\le }_{1}y.$$

In [3], we established the precise form of the Ekeland variational principle on a${\mathcal{T}}_c$-complete RN-module. 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.If$G\subset F$is 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 exists$z\in G$such that the following are satisfied:

• (1) $\phi (z)\le \phi (x_0)-\parallel z-x_0\parallel$;

• (2) for each$x\in G$such that$x\ne z$, $\phi (x)\nleqq \phi (z)-\parallel x-z\parallel$, 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 $\parallel x-y\parallel \le \phi (x)-\phi (y)$.

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, $F\subset X$a${\mathcal{T}}_c$-closed subset with the countable concatenationproperty, and$z\in X\setminus F$. Let$r,R,\rho \in L_{++}^0(\mathcal{F})$with$0<r<R<\rho$on Ω, then thereexists$x_0\in {\partial }_c(F)$such that

$$\parallel x_0-z\parallel \le \rho$$

and

$$D(z,r,x_0)\cap F=\{x_0\},$$

where$R:=\bigwedge \{\parallel z-a\parallel :a\in F\}$, 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: $x_1\tilde{\le }{x}_2$ 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 $\phi (x)=(\rho +r){(R-r)}^{-1}\parallel x\parallel$, $\mathrm{\forall }x\in E$.

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

$${\tilde{I}}_A\cdot \phi (x)={\tilde{I}}_A\cdot (\rho +r){(R-r)}^{-1}\parallel x\parallel ={\tilde{I}}_A\cdot (\rho +r){(R-r)}^{-1}\parallel {\tilde{I}}_A\cdot x\parallel ={\tilde{I}}_A\cdot \phi ({\tilde{I}}_A\cdot x),\mathrm{\forall }x\in E,$$

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 $x_1\tilde{\le }{x}_2$. Then one can have $x_2\in D(0,r,x_1)$; thus we can suppose

$$x_2=(1-t)x_1+tv,$$

(1)

where $t\in L_{+}^0(\mathcal{F})$, $0\le t\le 1$, and $v\in B_0(r)$.

From (1), it follows that $\parallel x_2\parallel \le (1-t)\parallel x_1\parallel +t\parallel v\parallel$, which implies

$$t(\parallel x_1\parallel -\parallel v\parallel )\le \parallel x_1\parallel -\parallel x_2\parallel .$$

From $R-r\le \parallel x_1\parallel -\parallel v\parallel$, one can have

$$t\le (\parallel x_1\parallel -\parallel x_2\parallel ){(R-r)}^{-1}.$$

(2)

Thus by (2) we have

$$\begin{array}{rcl}\parallel x_1-x_2\parallel & =& t\parallel v-x_1\parallel \le t(\parallel v\parallel +\parallel x_1\parallel )\\ \le & t(r+\rho )\le (r+\rho ){(R-r)}^{-1}(\parallel x_1\parallel -\parallel x_2\parallel ),\end{array}$$

which implies $x_1{\le }_{\phi }{x}_2$, 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 $x=t{x}_0+(1-t)v$, where $v\in B_0(r)$. Thus we have

$$\parallel x\parallel =\parallel t{x}_0+(1-t)v\parallel \le t\parallel x_0\parallel +(1-t)\parallel v\parallel \le t\rho +(1-t)r<\rho$$

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.