# A Minimax Theorem forL-0-Valued Functions on Random Normed Modules

Authors: Shien Zhao; Yuan Zhao
Journal: Journal of Function Spaces and Applications
 (2013)

---

## Abstract

We generalize the well-known minimax theorems toL¯0-valued functions on random normed modules. We first give some basic properties of an L0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locally L0-convex topology. Then, we introduce the definition of random saddle points. Conditions for an L0-valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem of L¯0-valued functions in the framework of random normed modules and random conjugate spaces.


---

## Body

## 1. Introduction

The classical minimax theorem, which originated from game theory, is an important content of nonlinear analysis. It has been applied in many fields, such as optimization theory, different equations, and fixed point theory. The first mathematical formulation was established by Neumann in 1928 [1]. Since then, various generalizations of Neumann's minimax theorem have been given by several scholars; see [2–7]. The classical minimax theorems for extended real-valued functions L:A×B→R- show that, under some suitable conditions of compactness, convexity, and continuity, the equality
(1)infy∈Bsupx∈Af(x,y)=supx∈Ainfy∈Bf(x,y)
holds. In 1980s, to meet the needs of vectorial optimization, minimax problems in this more general setting have been investigated; see [4–7]. In this paper, we generalize the well-known minimax theorems to L-0-valued functions on random normed modules (briefly RN modules).Random metric theory is based on the idea of randomizing the classical space theory of functional analysis. All the basic notions such asRN modules and random inner product modules (briefly RIP modules) and random locally convex modules (briefly RLC modules) together with their random conjugate spaces were naturally presented by Guo in the course of the development of random functional analysis, (cf. [8–12]). In the last ten years, random metric theory and its applications in the theory of conditional risk measures have undergone a systematic and deep development. Especially after 2009, in [13] Guo gives the relations between the basic results currently available derived from the two kinds of topologies, namely, the (ε,λ)-topology and the locally L0-convex topology. In [14], Guo gives some basic results on L0-convex analysis together with some applications to conditional risk measures and studies the relations among the three kinds of conditional convex risk measures. Furthermore, in [15] Guo et al. establish a complete random convex analysis over RN modules and RLC modules by simultaneously considering the two kinds of topologies in order to provide a solid analytic foundation for the module approach to conditional risk measures. These results pave the way for further research of the theory of random convex analysis and conditional risk measures.Motivated by the recent applications of random metric theory to conditional risk measures [13, 16, 17], in this paper, we establish a minimax theorem for L-0-valued functions on random normed modules. Theorem 1, which is the main result of this paper, can be seen as a natural extension of the classical minimax theorems and has potential applications in the further study of conditional risk measures.To introduce the main result of this paper, let us first recall some notation and terminology as follows:K: the scalar field R of real numbers or C of complex numbers;(Ω,ℱ,P): a probability space;L0(ℱ, K)= the algebra of equivalence classes of K-valued ℱ-measurable random variables on (Ω, ℱ, P);L0(ℱ)=L0(ℱ, R);L-0(ℱ)= the set of equivalence classes of extended real-valued ℱ-measurable random variables on (Ω, ℱ, P).Theorem 1.
Let(E,∥·∥) be a random strictly convex and random reflexive random normed module over R with base (Ω,ℱ,P), A and B  𝒯c-closed, L0(ℱ)-convex subset with the countable concatenation property of E and L:E×E→L-0(ℱ). If L satisfies the following:(1)
for any fixedp∈B, L(·,p):E→L-0(ℱ) is proper L0(ℱ)-convex, 𝒯c-lower semicontinuous function on A and has the local property;(2)
for any fixedu∈A, -L(u,·):E→L-0(ℱ) is proper L0(ℱ)-convex, 𝒯c-lower semicontinuous function on B and has the local property;(3)
A and B are a.s. bounded,then there exists a random saddle point (u0,p0)∈A×B of L with respect to A×B, namely,
(2)⋀u∈A⋁p∈BL(u,p)=L(u0,p0)=⋁p∈B⋀u∈AL(u,p).Theorem1 has the same shape as the classical minimax theorems, and its proof follows a known pattern in [18], but it is not trivial since the complicated stratification structure in the random setting needs to be considered. Besides, the most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. In order to overcome this obstacle, we make full use of the respective advantages of the (ε,λ)-topology and the locally L0-convex topology. In [13], Guo pointed out that these two kinds of topologies can complement each other (see also Propositions 14 and 15 in this paper), and we can consider them simultaneously in some cases. Specifically, on one hand, in Theorem 1 we require the functions to be 𝒯c-lower semicontinuous; namely, the functions are lower semicontinuous under the locally L0-convex topology, because we need a very important inequality to establish this theorem; see Definition 21 and Proposition 22 of this paper for details. On the other hand, in the process of the proof of Theorem 1 we must employ the (ε,λ)-topology also, because the (ε,λ)-topology is very natural from the viewpoint of probability theory, and under this type of topology we can use the relations between random normed modules and classical normed spaces to prove the main result; see the proof of Theorem 1 in Section 4.The remainder of this paper is organized as follows: in Section2 we will briefly collect some necessary known facts; in Section 3 we will give some basic properties of an L-0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, Theorems 26 and 28; in Section 4 we will present the definition of random saddle points and prove our main result.


## 2. Preliminaries

It is well known from [19] that L-0(ℱ) is a complete lattice under the ordering ≤: ξ≤η if and only if ξ0(ω)≤η0(ω), for almost all ω in Ω (briefly, a.s.), where ξ0 and η0 are arbitrarily chosen representatives of ξ and η, respectively. Furthermore, every subset G of L-0(ℱ) has a supremum, denoted by ⋁G, and an infimum, denoted by ⋀G. Finally L0(ℱ), as a sublattice of L-0(ℱ), is also a complete lattice in the sense that every subset with upper bound has a supremum. The pleasant properties of L-0(ℱ) are summarized as follows.Proposition 2 (see [19]).
For every subsetG of L-0(ℱ), there exist countable subsets {an∣n∈N} and {bn∣n∈N} of G such that ⋁G=⋁n≥1an and ⋀G=⋀n≥1bn. Further, if G is directed (dually directed) with respect to ≤, then the above {an∣n∈N}(accordingly, {bn∣n∈N}) can be chosen as nondecreasing (correspondingly, nonincreasing) with respect to ≤.Specially,L+0={ξ∈L0(ℱ)∣ξ≥0}, L++0={ξ∈L0(ℱ)∣ξ>0 on Ω}, where for A∈ℱ, “ξ>η” on A means ξ0(ω)>η0(ω) a.s. on A for any chosen representatives ξ0 and η0 of ξ and η, respectively. As usual, ξ>η means ξ≥η and ξ≠η. For any A∈ℱ, Ac denotes the complement of A, and A~={B∈ℱ∣P(AΔB)=0} denotes the equivalence class of A, where Δ is the symmetric difference operation, IA is the characteristic function of A, and I~A is used to denote the equivalence class of IA; given two ξ and η in L0(ℱ), and A={ω∈Ω∣ξ0≠η0}, where ξ0 and η0 are arbitrarily chosen representatives of ξ and η respectively, then we always write [ξ≠η] for the equivalence class of A and I[ξ≠η] for I~A; one can also understand the implication of such notation as I[ξ≤η], I[ξ<η] and I[ξ=η].For an arbitrarily chosen representativeξ0 of ξ∈L0(ℱ,K), define the two random variables (ξ0)-1 and |ξ0| by (ξ0)-1(ω)=1/ξ0(ω) if ξ0(ω)≠0, and (ξ0)-1(ω)=0 otherwise, and by |ξ0|(ω)=|ξ0(ω)|, for all ω∈Ω. Then the equivalent class ξ-1 of (ξ0)-1 is called the generalized inverse of ξ, and the equivalent class |ξ| of |ξ0| is called the absolute value of ξ.Now, we introduce the definition of a random normed module, which is a random generalization of an ordinary normed space, and give some important examples.Definition 3 (see [11, 20]).
An ordered pair(E,∥·∥) is called a random normed space (briefly, an RN space) over K with base (Ω,ℱ,P) if E is a linear space over K, and ∥·∥ is a mapping from E to L+0(ℱ) such that the following are satisfied:(RN-1)
∥αx∥=|α|∥x∥, for all α∈K and x∈E;(RN-2)
∥x∥=0 implies x=θ (the null element of E);(RN-3)
∥x+y∥≤∥x∥+∥y∥, for all x,y∈E.Here ∥·∥ is called the random norm on E and ∥x∥ the random norm of x∈E (if ∥·∥ only satisfies (RN-1) and (RN-3) above, it is called a random seminorm on E).
Furthermore, if, in addition,E is a left module over the algebra L0(ℱ,K) (briefly, an L0(ℱ,K)-module) such that(RNM-1)
∥ξx∥=|ξ|∥x∥, for all ξ∈L0(ℱ,K) and x∈E,then (E,∥·∥) is called a random normed module (briefly, an RN module) over K with base (Ω,ℱ,P), and the random norm ∥·∥ with the property (RNM-1) is also called an L0-norm on E (a mapping only satisfying (RN-3) and (RNM-1) above is called an L0-seminorm on E).Example 4.
LetL0(ℱ,B) be the L0(ℱ,K)-module of equivalence classes of ℱ-random variables (or strongly ℱ-measurable functions) from (Ω,ℱ,P) to a normed space (B,∥·∥) over K. ∥·∥ induces an L0-norm (still denoted by ∥·∥) on L0(ℱ,B) by ∥x∥:= the equivalence class of ∥x0(·)∥ for all x∈L0(ℱ,B), where x0(·) is a representative of x. Then (L0(ℱ,B),∥·∥) is an RN module over K with base (Ω,ℱ,P). Specially, L0(ℱ,K) is an RN module, and the L0-norm ∥·∥ on L0(ℱ,K) is still denoted by |·|.The next example ofRN modules Lℱp(ℰ)(1≤p≤+∞) is constructed by Filipović et al. in [16].Example 5.
Let(Ω,ℰ,P) be a probability space and ℱ a σ-subalgebra of ℰ. Define |||·|||p:L0(ℰ)→L-+0(ℱ) by
(3)|||x|||p={E[|x|p∣ℱ]1/p,when1≤p<∞,⋀{ξ∈L-+0(ℱ)∣|x|≤ξ},whenp=+∞,
for all x∈L0(ℰ).
DenoteLℱp(ℰ)={x∈L0(ℰ)|||x|||p∈L+0(ℱ)}, then (Lℱp(ℰ),|||·|||p) is an RN module over R with base (Ω,ℱ,P) and Lℱp(ℰ)=L0(ℱ)·Lp(ℰ)={ξx∣ξ∈L0(ℱ) and x∈Lp(ℰ)}.To put some important classes of stochastic processes into the framework ofRN modules, Guo constructed a more general RN module Lℱp(S) in [13] for each p∈[1,+∞] as follows.Example 6.
Let(E,∥·∥) be an RN module over K with base (Ω,ℰ,P) and ℱ a σ-subalgebra. Define |||·|||p:E→L-+0(ℱ) by
(4)|||·|||p={E[∥x∥p∣ℱ]1/p,when1≤p<∞,⋀{ξ∈L-+0(ℱ)∣∥x∥≤ξ},whenp=+∞,
for all x∈E.
DenoteLℱp(E)={x∈S∣|||x|||p∈L+0(ℱ)}; then (Lℱp(E),|||·|||p) is an RN module over K with base (Ω,ℱ,P). When E=L0(ℰ), Lℱp(E) is exactly Lℱp(ℰ).Remark 7.
For a givenRN module (E,∥·∥) over K with base (Ω,F,P) and a given real or extended real number p such that 1≤p≤+∞, define ∥·∥p:E→[0,+∞] by
(5)∥g∥p={(∫Ω(∥g∥)p‍)1/p,if1≤p<+∞,theP-essentialsupremum,ifp=+∞.
Let Lp(E)={g∈E:∥g∥p<+∞}. As mentioned in [13], (Lp(E),∥·∥p) is a normed space over K and is further a Banach space if (E,∥·∥) is complete.For eachRN module (E,∥·∥) over K with base (Ω,ℱ,P), ∥·∥ can induce two kinds of topologies, namely, the (ε,λ)-topology and the locally L0-convex topology.Definition 8 (see [12–14]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P). For any positive real numbers ε and λ such that 0<λ<1, let Nθ(ε,λ)={x∈E∣P{ω∈Ω∣∥x∥(ω)<ε}>1-λ}; then {Nθ(ε,λ)∣ε>0, 0<λ<1} is easily verified to be a local base at the null vector θ of some Hausdorff linear topology. The linear topology is called the (ε,λ)-topology for E induced by ∥·∥.From now on, the(ε,λ)-topology for each RN module is always denoted by 𝒯ε,λ when no confusion occurs.Proposition 9 (see [12–14]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P). Then one has the following statements.(1)
The(ε,λ)-topology forL0(ℱ,K)is exactly the topology of convergence in probabilityP, and(L0(ℱ,K),𝒯ε,λ)is a topological algebra overK.(2)
If(E,∥·∥)is anRNmodules, then(E,𝒯ε,λ)is a topological module over the topological algebraL0(F,K).(3)
A net{xδ,δ∈Γ}converges in the(ε,λ)-topology to somexinEif and only if{∥xδ-x∥,δ∈Γ}converges in probabilityPto
0.The following locallyL0-convex topology is easily seen to be much stronger than the (ε,λ)-topology and was first introduced by Filipović et al. in [16].Definition 10 (see [14, 16]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P). For any ε∈L++0, let Nθ(ε)={x∈E∣∥x∥≤ε}. A subset G of E is called 𝒯c-open if for each x∈G there exists some Nθ(ε) such that x+Nθ(ε)⊂G, and 𝒯c denotes the family of 𝒯c-open subsets of E. Then it is easy to see that (E,𝒯c) is a Hausdorff topological group with respect to the addition on E. 𝒯c is called the locally L0-convex topology for E induced by ∥·∥.From now on, the locallyL0-convex topology for each random locally convex space is always denoted by 𝒯c when no confusion occurs.Proposition 11 (see [13, 14, 16]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P). Then(1)
L0(ℱ,K)is a topological ring endowed with its locallyL0-convex topology;(2)
Eis a topological module over the topological ringL0(ℱ,K)whenEandL0(ℱ,K)are endowed with their respective locallyL0-convex topologies;(3)
a net{xα∣α∈Γ}inEconverges in the locallyL0-convex topology tox∈Eif and only if{∥xα-x∥∣α∈Γ}converges in the locallyL0-convex topology ofL0(ℱ,K)to
0.𝒯
c is called locally L0-convex because it has a striking local base 𝒰θ={B(ε)∣ε∈L++0}, each member U of which is as follows:(i)
L0-convex: ξ·x+(1-ξ)·y∈U for any x,y∈U and ξ∈L+0 such that 0⩽ξ⩽1;(ii)
L0-absorbent: there is ξ∈L++0 for each x∈E such that x∈ξ·U;(iii)
L0-balanced: ξ·x∈U for any x∈U and any ξ∈L0(F,K) such that |ξ|⩽1.Remark 12.
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P) endowed with the locally L0-convex topology 𝒯c. Although E can be viewed as a linear space over K with scalar multiplication α·x:=(α·1)·x for α∈K, 1∈L0 and x∈E, (E,𝒯c) is not a topological linear space since the map K→(E,𝒯c), α→α·x, is not necessarily continuous for x≠θ; see [16] for details.In the sequel of this paper, for a subsetG of an RN module (E,∥·∥), G-ε,λ denotes the 𝒯ε,λ-closure of G, and G-c denotes the 𝒯c-closure of G.For giving the relations of the two kinds of topologies, which Guo has studied the [13], we need to introduce the definition of the countable concatenation property.Definition 13 (see [13]).
LetE be a left module over the algebra L0(ℱ,K). A formal sum Σn≥1I~Anxn for some countable partition {An,n∈N} of Ω to ℱ and some sequence {xn∣n∈N} in E is called a countable concatenation of {xn∣n∈N} with respect to {An,n∈N}. Furthermore a countable concatenation Σn≥1I~Anxn is well defined or Σn≥1I~Anxn∈E if there is x∈E such that I~Anx=I~Anxn, for all n∈N. A subset G of E is said to have the countable concatenation property if every countable concatenation Σn≥1I~Anxn with xn∈G for each n∈N still belongs to G; namely, Σn≥1I~Anxn is well defined and there exists x∈G such that x=Σn≥1I~Anxn.Proposition 14 (see [13]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P). Then E is 𝒯ε,λ-complete if and only if E is 𝒯c-complete and has the countable concatenation property.Proposition 15 (see [13]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P) and G⊂E a subset with the countable concatenation property. Then G-ε,λ=G-c.Now, we introduce the definition of random conjugate spaces ofRN modules.Definition 16 (see [7, 10, 11, 13]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P). A linear operator f from E to L0(ℱ,K) is said to be an a.s. bounded random linear functional on E if there exists some ξ in L+0(ℱ,R) such that |f(x)|≤ξ·∥x∥, for all x∈E. Denote by E* the linear space of a.s. bounded random linear functionals on E with the pointwise addition and scalar multiplication on linear operators; define ∥·∥*:E*→L+0(ℱ,R) by ∥f∥*=⋀{ξ∈L+0(ℱ)∣|f(x)|≤ξ∥x∥,forallx∈E} for all f∈E* and define ·:L0(ℱ,K)×E*→E* by (η·f)(x)=η(f(x)) for all η∈L0(ℱ,K),  f∈E*, and x∈E; then it is easy to check that (E*,∥·∥*) is also an RN module over K with base (Ω,ℱ,P), called the random conjugate space of (E,∥·∥).Guo et al. gave the topological characterizations of an a.s. bounded random linear functional in [10, 11, 16] as follows: let (E,∥·∥) be an RN module over K with base (Ω,ℱ,P), Eε,λ* the L0(ℱ,K)-module of continuous module homomorphisms from (E,𝒯ε,λ) to (L0(ℱ,K),𝒯ε,λ), and Ec* the L0(F,K)-module of continuous module homomorphisms from (E,𝒯c) to (L0(F,K),𝒯c), then it was proved that Eε,λ*=Ec*. In fact, Guo et al. also proved in [10, 11, 13] ∥f∥*=⋁{|f(x)|∣x∈Eand∥x∥≤1} for any f∈E.Let(E,∥·∥) be an RN module, E** denotes (E*)*, and the canonical embedding mapping J:E→E** defined by (Jx)(f)=f(x), for all x∈E and for all f∈E*, is random-norm preserving. If J is subjective, then E is called random reflexive. In [13] Guo proved that the random reflexivity is independent of a special choice of 𝒯ε,λ and 𝒯c. The following propositions are very essential relations, which are established by Guo in [12, 21], between classical reflexive spaces and random reflexive RN modules.Proposition 17 (see [21]).
L
0
(
ℱ
,
B
) is random reflexive if and only if B is a reflexive Banach space.Proposition 18 (see [12]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P). Then E is random reflexive if and only if (Lp(E),∥·∥p) is reflexive, where 1<p<+∞.Proposition 19 (see [12]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P), 1≤p<+∞ and 1<q≤+∞ a pair of Hölder conjugate numbers. Then (Lq(E*),∥·∥q) is isometrically isomorphic with the classical conjugate space of (Lp(E),∥·∥), denoted by (Lp(E))′, under the canonical mapping T:Lq(E*)→(Lp(E))′ defined as follows. For each f∈Lq(E*), Tf (denoting T(f)):Lp(E)→K is defined by Tf(g)=∫Ω‍f(g)dP for all g∈Lp(E).


## 3. Some Basic Properties ofL-0-Valued Lower Semicontinuous Functions

In this section, we give some basic properties ofL-0-valued lower semicontinuous functions. First, we recall the definition of L-0-valued lower semicontinuous functions under two kinds of topologies, which was presented by Guo in [14] for the first time.LetE be a left module over the algebra L0(ℱ). The effective domain of function f:E→L-0(ℱ) is denoted by dom(f):={x∈E∣f(x)∈L0(ℱ)}. The epigraph of f is denoted by epi(f):={(x,y)∈E×L0(ℱ)∣f(x)≤y}. The function f is called proper if f(x)>-∞ on Ω for every x∈E and dom(f)≠∅.Definition 20 (see [14]).
LetE be a left module over the algebra L0(ℱ) and f:E→L-0(ℱ).(1)
f is L0(ℱ)-convex if f(ξx+(1-ξ)y)≤ξf(x)+(1-ξ)f(y) for all x and y in E and ξ∈L+0 such that 0≤ξ≤1 (here we make the convention that 0·(±∞)=0 and ∞-∞=∞).(2)
f has the local property if I~Af(x)=I~Af(I~Ax) for all x∈E and A∈ℱ.(3)
f is regular if I~Af(x)=f(I~Ax) for all x∈E and A∈ℱ.Definition 21 (see [14]).
Let(E,∥·∥) be an RN module over R with base (Ω,ℱ,P). A function f:E→L-0(ℱ) is called 𝒯c-lower semicontinuous if epi(f) is closed in (E,𝒯c)×(L0(ℱ),𝒯c). A function f:E→L-0(ℱ) is called 𝒯ε,λ-lower semicontinuous if epi(f) is closed in (E,𝒯ε,λ)×(L0(ℱ),𝒯ε,λ).Proposition 22 (see [14]).
Let(E,∥·∥) be an RN module over R with base (Ω,ℱ,P) such that E has the countable concatenation property and f:E→L-0(ℱ) a function with the local property. Then the following are equivalent to each other:(1)
fis𝒯c-lower semicontinuous;(2)
{x∈E∣f(x)≤r}is𝒯c-closed for eachr∈L0(ℱ);(3)
lim_αf(xα)≥f(x0)for eachx0∈Eand each net{xα,α∈Γ}inEsuch that{xα,α∈Γ}is𝒯c-convergent tox0, wherelim_αf(xα)=⋁α∈Γ(⋀β≥αf(xβ)).Remark 23.
Proposition22 first occurred in [16] where the countable concatenation property of E was not assumed, but this condition should be added (see [14] for details).For𝒯ε,λ-lower semicontinuous functions, we only have the following proposition.Proposition 24 (see [14]).
Let(E,∥·∥) be an RN module over R with base (Ω,ℱ,P) and f:E→L-0(ℱ) a function. Then one has the following statements:(1)
fis𝒯ε,λ-lower semicontinuous iflim_αf(xα)≥f(x0)for eachx0∈Eand each net{xα,x∈Γ}inEsuch that{xα,α∈Γ}is𝒯ε,λ-convergent tox0;(2)
{x∈E∣f(x)≤r}is𝒯ε,λ-closed for eachr∈L0(ℱ)iffis𝒯ε,λ-lower semicontinuous.If we definef to be lower semicontinuous via “lim_αf(xα)≥f(x) for all net {xα,α∈Λ} in E such that it converges in the (ε,λ)-topology to some x∈E”, the notion is, however, meaningless in the random setting, since we can construct an RN module E and a 𝒯ε,λ-continuous L0-convex function f from E to L0(ℱ), whereas f is not a 𝒯ε,λ-lower semicontinuous function. Hence, we cannot use this inequality for 𝒯ε,λ-lower semicontinuous functions. Since this inequality is very important for the proof of Theorem 1 (see Section 4 for details), we can only establish L-0-valued minimax theorems for 𝒯c-lower semicontinuous functions.Proposition 25 (see [14]).
Let(E,∥·∥) be an RN module over K with base (Ω,ℱ,P) such that E has the countable concatenation property and f:E→L-0(ℱ) a function with the local property. Then f is 𝒯ε,λ-lower semicontinuous if and only if f is 𝒯c-lower semicontinuous, specially this is true for an L0(ℱ)-convex function f.Now, we give some important properties ofL-0-valued lower semicontinuous functions on RN modules. To pave the way for Theorem 26, we first introduce some notation: if (E,∥·∥) is an RN module over K with base (Ω,ℱ,P), Π denotes the set of all probability measures equivalent to P on (Ω,ℱ), LQp(E)={x∈E∣∫Ω‍∥x∥pdQ<+∞}, where Q∈Π, and ∥·∥pQ denotes the norm on LQp(E), namely, ∥x∥pQ=(∫Ω‍∥x∥pdQ)1/p for any x∈LQp(E).Theorem 26.
Let(E,∥·∥) be a random reflexive RN module over R with base (Ω,ℱ,P), G⊂E a 𝒯c-closed, L0(ℱ)-convex, and a.s. bounded set with the countable concatenation property and f:E→L-0(ℱ) a 𝒯c-lower semicontinuous function with the local property. If f∣G is proper, then f is bounded from below on G.Proof.
Letη=∨{∥x∥:x∈G}; then it is easy to see that η∈L+0. If f is not bounded from below on G, then there exists B∈ℱ such that P(B)>0 and
(6)⋀{f(x)∣x∈G}=-∞
on B. Since G is L0(ℱ)-convex and f has the local property, it is easy to see that {f(x)∣x∈G} is directed. Hence there exists a sequence {xn,n∈N} such that {f(xn),n∈N}↘⋀{f(x)∣x∈G}. Let Gn={x∈E∣f(x)≤f(xn)}⋂G; then it is clear that Gn is 𝒯c-closed by Definition 21. Since f has the local property and G has the countable concatenation property, for any n∈N, we have that Gn has the countable concatenation property and Gn is 𝒯ε,λ-closed by Proposition 15. By the fact that G is a.s. bounded, we can define a probability measure Q on (Ω,ℱ) by dQ/dP=1/c(1+η)2, where c=E[1/(1+η)2]. Then Q is equivalent to P and ∫Ω‍∥x∥2dQ≤∫Ω‍(η2/c(1+η)2)dP<+∞, for any x∈G, which means that G is bounded in (LQ2(E),∥·∥2Q). Noting that replacing the probability measure P of the base space (Ω,ℱ,P) with a probability measure Q does not change the (ε,λ)-topology of E, for any given n∈N, we can obtain that Gn is norm-closed and convex in (LQ2(E),∥·∥2Q). Since (E,∥·∥) is a random reflexive RN module, we have that (LQ2(E),∥·∥2Q) is reflexive normed space from Proposition 18. Hence Gn is compact under the weak topology of the normed space LQ2(E). Let 𝒪={Gn∣n∈N}, then one can obtain that 𝒪 has the finite intersection property and ⋂𝒪≠∅. Let x*∈⋂𝒪; then
(7)f(x*)=-∞
on B, which it contradicts to the fact that f|G is proper.For giving Theorem28, we need to introduce the following Proposition 27, which was established by Guo and Yang in [22] for studying Ekeland's variational principle for L-0-valued functions on RN modules.Proposition 27 (see [22]).
Let(E,∥·∥) be an RN module over R with base (Ω,ℱ,P), G⊂E a subset with the countable concatenation property and f:E→L-0(ℱ) have the local property. If f|G is proper and bounded from below on G (resp., bounded from above on G), then for each ε∈L++0(ℱ), there exists xε∈G such that f(xε)≤⋀f(G)+ε (accordingly, f(xε)≥⋁f(G)-ε).Theorem 28.
Let(E,∥·∥) be a random reflexive RN module over R with base (Ω,ℱ,P), G⊂E a 𝒯c-closed, L0(ℱ)-convex and a.s. bounded set with the countable concatenation property and f:E→L-0(ℱ) a 𝒯c-lower semicontinuous and L0(ℱ)-convex function with the local property. If f|G is proper, then there exists x*∈G such that f(x*)=⋀f(G).Proof.
Letξ=⋁{∥x∥∣x∈G} and η=⋀f(G). It is clear that ξ∈L+0 and η∈L0(ℱ) by Theorem 26. Take Bj=[j-1≤ξ<j], for all j∈N, then {Bj∣j=1,2,…} is a countable partition of Ω to ℱ. For any j∈N; define a function fj:I~Bj·E→L-0(ℱ) as follows:
(8)fj(I~Bjx)=I~Bjf(x),∀x∈E.
Since f has the local property, for any j∈N, we have that fj(I~Bjx)=I~Bjf(I~Bjx) and I~Bj·⋀f(G)=⋀fj(I~Bj·G). Because f|G is proper and f is 𝒯c-lower semicontinuous and L0(ℱ)-convex, it is clear that fj|I~Bj·G is proper, 𝒯c-lower semicontinuous, and L0(ℱ)-convex. Next, we prove that fj has the local property. We need only to prove that
(9)I~Afj(I~AI~Bjx)=I~Afj(I~Bjx),∀A∈ℱ⋂Bj.
In fact, since f has the local property, for any A∈ℱ⋂Bj, we have that
(10)I~Afj(I~Bjx)=I~AI~Bjf(x)=I~AI~Bjf(I~Bjx)=I~AI~Bjf(I~AI~Bjx)=I~Afj(I~AI~Bjx).
Let ηj=I~Bj·η for any j∈N. It is easy to see that I~jG⊆(L2(E),∥·∥2) is a bounded and convex set. Since I~BjG is L0(ℱ)-convex, 𝒯c-closed, and a.s. bounded in E and has the countable concatenation, we can obtain that I~BjG is 𝒯ε,λ-closed in E by Proposition 15. It is easy to see that I~BjG is convex and ∥·∥2-closed in (L2(E),∥·∥2) from the fact that the topology induced by ∥·∥2 is stronger than the (ε,λ)-topology. Since (E,∥·∥) is random reflexive, (L2(E),∥·∥2) is reflexive normed space, and I~BjG is compact in L2(E) under the weak topology of L2(E). For any ε∈L++0, define Gj(ε)={I~Bjx∣fj(I~Bjx)≤I~Bjη+ε,x∈G}. It is clear that Gj(ε)≠∅ by Proposition 27. Since fj|I~Bj·G is 𝒯c-lower semicontinuous, we have that Gj(ε) is 𝒯c-closed. Thus, we have that Gj(ε) is L0(ℱ)-convex and ∥·∥2-closed in L2(E) by the fact that fj has the local property and Proposition 15. By Hahn-Banach theorem, we have that Gj(ε) is closed under the weak topology of L2(E). Take
(11)𝒪={Gj(ε)∣ε∈L++0},
it is easy to prove that 𝒪 has the finite intersection property. Since I~BjG is compact under the weak topology of L2(E), we have that ⋂𝒪≠∅. Let xj∈⋂𝒪 for any j∈N and
(12)x*=∑j=1∞‍I~Bj·xj.
We have that fj(xj)=ηj and
(13)f(x*)=⋀f(G).
This completes the proof.Definition 29.
Let(E,∥·∥) be an RN module over R with base (Ω,ℱ,P), and G a L0(ℱ)-convex subset in E. f:G→L-0(ℱ) is called strictly L0(ℱ)-convex if
(14)f(αx+(1-α)y)<αf(x)+(1-α)f(y),
for any x,y∈G, x≠y and 0<α<1 on Ω.Corollary 30.
Let(E,∥·∥) be a random reflexive RN module over R with base (Ω,ℱ,P), G⊂E a 𝒯c-closed, L0(ℱ)-convex and a.s. bounded set with the countable concatenation property, and f:E→L-0(ℱ) a 𝒯c-lower semicontinuous and strictly L0(ℱ)-convex function with the local property. If f|G is proper, then there exists an unique x*∈G such that f(x*)=⋀f(G).


## 4. Main Results

Now, we give the definition of random saddle points.Definition 31.
LetA and B be any two nonempty sets, (u0,p0)∈A×B and L:A×B→L-0(ℱ). Then (u0,p0) is called a random saddle point of f with respect to A×B if
(15)L(u0,p)≤L(u0,p0)≤L(u,p0),∀u∈A,p∈B.Remark 32.
LetA and B be any two nonempty sets, L:A×B→L-0(ℱ). It is easy to see that the following statements are equivalent:(1)
(u0,p0)∈A×B is a random saddle point of f with respect to A×B;(2)
⋀u∈A⋁p∈BL(u,p)≤⋁p∈B⋀u∈AL(u,p);(3)
⋀u∈A⋁p∈BL(u,p)=L(u0,p0)=⋁p∈B⋀u∈AL(u,p).Before giving the proof of main result in this paper, we first recall the definition of random strictly convexRN module, which is presented by Guo and Zeng in [23] for the first time.Definition 33 (see [23]).
AnRN module (E,∥·∥) is said to be random strictly convex if for any x and y∈E∖{θ} such that ∥x+y∥=∥x∥+∥y∥, then there exist A∈ℱ and ξ∈L+0 such that P(A)>0, ξ>0 on A and I~Ax=ξ(I~Ay).Definition 34 (see [24]).
Let(E,∥·∥) be an RN module over R with base (Ω,ℱ,P), x,y∈E and F∈ℱ. Then x and y are called L0-independent on F if ξI~F=ηI~F=0 whenever ξ,η∈L0(ℱ) such that ξI~Fx+ηI~Fy=θ.By Definitions33 and 34, we can obtain the following lemma easily.Lemma 35.
Let(E,∥·∥) be a random strictly convex RN module over R with base (Ω,ℱ,P). Then the mapping f:E→L0(ℱ)(16)f(x)=∥x∥2
is strictly L0(ℱ)-convex.For giving the proof of Theorem1, we need the following lemma and remark.Lemma 36 (Mazur lemma).
Let(X,∥·∥) be a normed space, {un∈X,n∈N} converge to u- under the weak topology on X. Then there exists a sequence {vn∈X,n∈N} such that it converges to u- in norm, where
(17)vn=∑k=nNn‍λkuk,∑k=nNn‍λk=1,λk≥0.Remark 37.
Let{ξn∈L0(ℱ),n∈N} converge to ξ uniformly. Then we can obtain a net {ξε∈L0(ℱ)∣ε∈L++0,ε≤1} such that it converges to ξ under the locally L0-convex topology of (L0(ℱ,R),|·|). In fact, for any ε∈L++0 let A1=[ε>1],Ai=[1/(i+1)<ε≤1/i], for all i∈N. Then {An,n∈N} is a countable partition of Ω to ℱ. Since the sequence {ξn,n∈N} converges to ξ uniformly, thus for any number k>0, there exists N(k)∈N such that
(18)|ξn-ξ|<1k
for any n>N(k). Let
(19)ξε=∑i=1∞‍I~AiξN(i)+1;
then it is easy to see that |ξε-ξ|<ε. Set ∧={ε∈L++0∣ε≤1}; then ∧ is directed with respect to ≤, and one can easy to see that the net {ξε,ε∈∧} converges to ξ under the locally L0-convex topology of (L0(ℱ),|·|).With the above preparations, we now give the proof of Theorem1.Proof of Theorem1.
First, let us assume that for anyp∈B, L(·,p) is strictly L0(ℱ)-convex on A. Set F(u)=⋁p∈BL(u,p) and G(p)=⋀u∈AL(u,p). We show that the functional F has the local property and F is L0(ℱ)-convex and 𝒯c-lower semicontinuous on A. By conditions (1) and (2), it is clear that F has the local property. For any x1,x2∈A,α∈L+0, we have that
(20)F(αx1+(1-α)x2)=⋁p∈BL(αx1+(1-α)x2,p)≤⋁p∈B[αL(x1,p)+(1-α)L(x2,p)]≤α⋁p∈BL(x1,p)+(1-α)⋁p∈BL(x2,p)=αF(x1)+(1-α)F(x2).
Thus, F is L0(ℱ)-convex on A. For any r∈L0(ℱ,R), let Ar={u∈A∣F(u)≤r}. Since F has the local property, we have that Ar has the countable concatenation property. Let {xα,α∈∧}⊂Ar converge to x0 under the locally L0-convex topology of E. By F(xα)≤r, we can obtain that
(21)L(xα,p)≤r,∀p∈B.
Since L(·,p) is 𝒯c-lower semicontinuous, it is easy to see that
(22)L(x0,p)≤r,∀p∈B.
Hence, F(x0)≤r and x0∈Ar. So we have that F is 𝒯c-lower semicontinuous. Similarly, replacing L with -L, we see that -G has the local property and -G is L0(ℱ)-convex and 𝒯c-lower semicontinuous on B.
By Theorems26 and 28, we have that ⋁p∈BG(p)∈L0(ℱ), and there exists p0∈B such that
(23)G(p0)=⋁p∈BG(p).
Since for all p∈B, L(·,p) is strictly L0(ℱ)-convex on A, according to Corollary 30 there exists an unique up∈A such that
(24)G(p)=⋀u∈AL(u,p)=L(up,p),
for any p∈B. Let u0=up0, we need only to prove that
(25)G(p0)≥L(u0,p),∀p∈B.
For everyp∈B, let pn=(1-n-1)p0+n-1p and un=upn,  for all n∈N. It is clear that
(26)G(p0)≥G(pn)=L(un,pn).
By condition (2), we have that
(27)G(p0)≥L(un,pn)≥(1-n-1)L(un,p0)+n-1L(un,p),G(p0)≥(1-n-1)G(p0)+n-1L(un,p),
namely, G(p0)≥L(un,p). Let η=⋁{∥x∥∣x∈A} and Cn={ω∈Ω∣n-1≤η<n}, for all n∈N, according to the condition that A is a.s. bounded in E, and it is easy to see that {Cn,n∈N} is a countable partition of Ω to ℱ.
For any fixedi∈N, we have that I~Ci·un∈L2(E), for all n∈N. It is easy to see that I~Ci·A is a bounded and closed subset of (L2(E),∥·∥2). Thus, there exists a subsequence {I~Ci·unk,k∈N} and ωi∈I~Ci·A such that {I~Ci·unk,k∈N} converges to ωi under the weak topology of L2(E). Without loss of generality, denote this subsequence by {I~Ci·un,n∈N}. By Lemma 36, there exists a sequence {vn∈I~Ci·A,n∈N} such that it converges to ωi in norm, where
(28)vn=∑k=nNn‍λkuk,∑k=nNn‍λk=1,λk≥0. Since ∥vn-ωi∥2→0, we have that {∥vn-ωi∥,n∈N} converges in probability P to θ. By Egoroff theorem, for any number δ>0, there exists Cδ∈ℱ such that P(Cδ)<δ and {∥vn-ωi∥,n∈N} converges uniformly to 0 on Cδc. Then there is a net
(29){∥vε-ωi∥,ε∈∧}
as in Remark 37, which converges 0 on Cδc under the locally L0-convex topology of (L0(ℱ),|·|). By the construction of {∥vε-ωi∥,ε∈∧} as in Remark 37, we have that
(30)lim_n→∞L(vn,p)≥lim_ε∈∧L(vε,p).
Since L(·,p) is L0(ℱ)-convex for any p∈B, we have that
(31)I~CδcI~CiG(p0)≥I~CδcI~Cilim_n→∞∑k=nNn‍λkL(uk,p)≥I~CδcI~Cilim_n→∞L(vn,p)≥I~CδcI~Cilim_ε∈∧L(vε,p)≥I~CδcI~CiL(ωi,p).
Hence, one can obtain that I~CδcI~CiG(p0)≥I~CδcI~CiL(ωi,p). Because δ is an arbitrary nonnegative number and L(·,p) has the local property, we have that
(32)I~CiG(p0)≥I~CiL(ωi,p),
for any p∈B.
Now, we prove thatωi=I~Ciu0 for any i∈N. By the definition of un, it is clear that
(33)L(un,pn)≤L(u,pn),∀u∈A.
By condition (2), we have that
(34)(1-n-1)L(un,p0)+n-1L(un,p)≤L(u,pn),∀p∈B.
Hence, we can obtainthat
(35)lim¯n→∞(1-n-1)L(un,p0)≤lim¯n→∞L(u,pn),∀p∈B.
Since L(un,p)≥G(p), we can obtain that (1-n-1)L(un,p0)+n-1G(p)≤L(u,pn) and (1-n-1)L(vn,p0)+n-1G(p)≤I~Ci·lim¯n→∞((1-n-1)L(un,p0)+n-1L(un,p)). According to
(36)lim_n→∞L(vn,p0)=lim_n→∞(1-n-1)L(vn,p0)≤lim_n→∞(1-n-1)∑k=nNn‍λkL(uk,p0)≤lim-n→∞(1-n-1)L(un,p0)≤lim-n→∞L(u,pn),
where ∑k=nNnλk=1, it is obvious that
(37)I~CiL(ωi,p0)≤lim_n→∞I~CiL(vn,p0)≤lim¯n→∞I~CiL(un,p0)≤lim¯n→∞I~CiL(u,pn).
Since {∥pn-p0∥,n∈N} converges in probability to 0, by Egoroff theorem, we can obtain that for any σ>0, there exists Cσ∈ℱ such that P(Cσ)<σ and {∥pn-p0∥,n∈N} converges to 0 uniformly on Cσc. By Remark 37, we can construct a net {pα∈B,α∈∧} as in Remark 37 such that ∥pα-p0∥→0 under the locally L0-convex topology of E. Hence, by (2) we can obtain that
(38)I~CσcI~CiL(ωi,p0)≤I~CσcI~Cilim¯α∈∧L(u,pα)≤I~CσcI~CiL(u,p0).
Since σ is an arbitrary nonnegative number, it is clear that
(39)I~CiL(ωi,p0)≤I~CiL(u,p0).
Therefore, for any i∈N, ωi=I~Ciu0 and G(p0)≥L(u0,p), for all p∈B; namely, (u0,p0) is a random saddle point of L with respect to A×B.
If there isp∈B such that L(·,p) is not strictly L0(ℱ)-convex on A, define
(40)Ln(u,p)=L(u,p)+n-1∥u∥2,∀n∈N.
Since E is random strictly convex RN module, we can obtain that Ln is strictly L0(ℱ)-convex from Lemma 35. By the similar method, we have that for any n∈N, there exists (un,pn)∈A×B such that it is a saddle point of Ln with respect to A×B.
Letζ=⋁{∥y∥∣y∈B} and Dm be any representation element of [m-1≤η<m], for all m∈N, according to the condition that B is a.s. bounded in E, and it is easy to see that {Dm,m∈N} is a countable partition of Ω to ℱ. It is easy to see that {Ci⋂Dj,i,j∈N} is also a countable partition of Ω to ℱ. For any i,j∈N, let Hij=Ci⋂Dj. We can suppose that, without loss of generality, I~Hijun∈L2(E) converge to μij under the weak topology of L2(E). Then we have that there exists a net {I~Hijuα∈E∣α∈Λ} such that it converges to μij under the locally L0-convex topology of E. Thus, we have that
(41)I~HijL(μij,p)≤lim_αI~HijL(I~Hijuα,p)≤lim_n→∞I~HijL(I~Hijun,p)+n-1∥un∥2≤lim¯n→∞(I~HijL(u,I~Hijpn)+n-1∥u∥2)
for all u∈A,p∈B. Similarly, for any i,j∈N, one can have a net {I~Hijpβ∈E∣β∈Γ} such that it converges to νij under the locally L0-convex topology of E and
(42)lim¯n→∞(I~HijL(u,I~Hijpn)+n-1∥u∥2)≤lim¯n→∞I~HijL(u,I~Hijpn)+lim¯n→∞n-1∥u∥2≤lim¯β∈ΓI~HijL(u,I~Hijpβ)+lim¯n→∞n-1∥u∥2≤I~HijL(u,νij).
Let u0=∑i,j∈NI~Hijμij and p0=∑i,j∈NI~Hijνij; it is easy to check that
(43)L(u0,p)≤L(u0,p0)≤L(u,p0)
for all u∈A,p∈B; namely, (u0,p0) is a random saddle point of L with respect to A×B.
This completes the proof.



---

*Source: 704251-2013-10-05.xml*