#### Abstract

We generalize the well-known minimax theorems to -valued functions on random normed modules. We first give some basic properties of an -valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the ()-topology and the locally -convex topology. Then, we introduce the definition of random saddle points. Conditions for an -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 -valued functions in the framework of random normed modules and random conjugate spaces.

#### 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 show that, under some suitable conditions of compactness, convexity, and continuity, the equality 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 -valued functions on random normed modules (briefly modules).

Random metric theory is based on the idea of randomizing the classical space theory of functional analysis. All the basic notions such as modules and random inner product modules (briefly modules) and random locally convex modules (briefly 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 -topology and the locally -convex topology. In [14], Guo gives some basic results on -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 modules and 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 -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: the scalar field R of real numbers or C of complex numbers; : a probability space; , the algebra of equivalence classes of -valued -measurable random variables on , , ; , ; the set of equivalence classes of extended real-valued -measurable random variables on , , .

Theorem 1. *Let be a random strictly convex and random reflexive random normed module over with base , and -closed, -convex subset with the countable concatenation property of and . If satisfies the following:*(1)*for any fixed , is proper -convex, -lower semicontinuous function on and has the local property;*(2)*for any fixed , is proper -convex, -lower semicontinuous function on and has the local property;*(3)* and are a.s. bounded,**then there exists a random saddle point of with respect to , namely,
*

Theorem 1 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 -topology and the locally -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 -lower semicontinuous; namely, the functions are lower semicontinuous under the locally -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 -topology also, because the -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 Section 2 we will briefly collect some necessary known facts; in Section 3 we will give some basic properties of an -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 is a complete lattice under the ordering : if and only if , for almost all in (briefly, a.s.), where and are arbitrarily chosen representatives of and , respectively. Furthermore, every subset of has a supremum, denoted by , and an infimum, denoted by . Finally , as a sublattice of , is also a complete lattice in the sense that every subset with upper bound has a supremum. The pleasant properties of are summarized as follows.

Proposition 2 (see [19]). *For every subset of , there exist countable subsets and of such that and . Further, if is directed (dually directed) with respect to , then the above (accordingly, ) can be chosen as nondecreasing (correspondingly, nonincreasing) with respect to . *

Specially, , on , where for , “” on means a.s. on for any chosen representatives and of and , respectively. As usual, means and . For any , denotes the complement of , and denotes the equivalence class of , where is the symmetric difference operation, is the characteristic function of , and is used to denote the equivalence class of ; given two and in , and , where and are arbitrarily chosen representatives of and respectively, then we always write for the equivalence class of and for ; one can also understand the implication of such notation as , and .

For an arbitrarily chosen representative of , define the two random variables and by if , and otherwise, and by , for all . Then the equivalent class of is called the generalized inverse of , and the equivalent class of is called the absolute value of .

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 is called a random normed space (briefly, an space) over with base if is a linear space over , and is a mapping from to such that the following are satisfied:(RN-1), for all and ;(RN-2) implies (the null element of );(RN-3), for all .Here is called the random norm on and the random norm of (if only satisfies (RN-1) and (RN-3) above, it is called a random seminorm on ).

Furthermore, if, in addition, is a left module over the algebra (briefly, an -module) such that(RNM-1), for all and ,then is called a random normed module (briefly, an module) over with base , and the random norm with the property (RNM-1) is also called an -norm on (a mapping only satisfying (RN-3) and (RNM-1) above is called an -seminorm on ).

*Example 4. *Let be the -module of equivalence classes of -random variables (or strongly -measurable functions) from to a normed space over . induces an -norm (still denoted by ) on by the equivalence class of for all , where is a representative of . Then is an module over with base . Specially, is an module, and the -norm on is still denoted by .

The next example of modules is constructed by Filipović et al. in [16].

*Example 5. *Let be a probability space and a -subalgebra of . Define by
for all .

Denote , then is an module over with base and and .

To put some important classes of stochastic processes into the framework of modules, Guo constructed a more general module in [13] for each as follows.

*Example 6. *Let be an module over with base and a -subalgebra. Define by
for all .

Denote ; then is an module over with base . When , is exactly .

*Remark 7. *For a given module over with base and a given real or extended real number such that , define by
Let . As mentioned in [13], is a normed space over and is further a Banach space if is complete.

For each module over with base , can induce two kinds of topologies, namely, the -topology and the locally -convex topology.

*Definition 8 (see [12–14]). *Let be an module over with base . For any positive real numbers and such that , let ; then , is easily verified to be a local base at the null vector of some Hausdorff linear topology. The linear topology is called the -topology for induced by .

From now on, the -topology for each module is always denoted by when no confusion occurs.

Proposition 9 (see [12–14]). *Let be an module over with base . Then one has the following statements.*(1)*The **-topology for ** is exactly the topology of convergence in probability **, and ** is a topological algebra over **.*(2)*If ** is an ** modules, then ** is a topological module over the topological algebra**.*(3)*A net ** converges in the **-topology to some ** in ** if and only if ** converges in probability ** to *
0.

The following locally -convex topology is easily seen to be much stronger than the -topology and was first introduced by Filipović et al. in [16].

*Definition 10 (see [14, 16]). *Let be an module over with base . For any , let . A subset of is called -open if for each there exists some such that , and denotes the family of -open subsets of . Then it is easy to see that is a Hausdorff topological group with respect to the addition on . is called the locally -convex topology for induced by .

From now on, the locally -convex topology for each random locally convex space is always denoted by when no confusion occurs.

Proposition 11 (see [13, 14, 16]). *Let be an module over with base . Then*(1)* is a topological ring endowed with its locally **-convex topology;*(2)* is a topological module over the topological ring ** when ** and ** are endowed with their respective locally **-convex topologies;*(3)*a net ** in ** converges in the locally **-convex topology to ** if and only if ** converges in the locally **-convex topology of ** to *
0.

is called locally -convex because it has a striking local base , each member of which is as follows:(i)-convex: for any and such that ;(ii)-absorbent: there is for each such that ;(iii)-balanced: for any and any such that .

*Remark 12. *Let be an module over with base endowed with the locally -convex topology . Although can be viewed as a linear space over with scalar multiplication for , and , is not a topological linear space since the map , , is not necessarily continuous for ; see [16] for details.

In the sequel of this paper, for a subset of an module , denotes the -closure of , and denotes the -closure of .

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]). *Let be a left module over the algebra . A formal sum for some countable partition of to and some sequence in is called a countable concatenation of with respect to . Furthermore a countable concatenation is well defined or if there is such that , for all . A subset of is said to have the countable concatenation property if every countable concatenation with for each still belongs to ; namely, is well defined and there exists such that .

Proposition 14 (see [13]). *Let be an module over with base . Then is -complete if and only if E is -complete and has the countable concatenation property.*

Proposition 15 (see [13]). *Let be an module over with base and a subset with the countable concatenation property. Then .*

Now, we introduce the definition of random conjugate spaces of modules.

*Definition 16 (see [7, 10, 11, 13]). *Let be an RN module over with base . A linear operator from to is said to be an a.s. bounded random linear functional on if there exists some in such that , for all . Denote by the linear space of a.s. bounded random linear functionals on with the pointwise addition and scalar multiplication on linear operators; define by for all and define by for all , , and ; then it is easy to check that is also an module over with base , called the random conjugate space of .

Guo et al. gave the topological characterizations of an a.s. bounded random linear functional in [10, 11, 16] as follows: let be an module over with base , the -module of continuous module homomorphisms from to , and the -module of continuous module homomorphisms from to , then it was proved that . In fact, Guo et al. also proved in [10, 11, 13] for any .

Let be an module, denotes , and the canonical embedding mapping defined by , for all and for all , is random-norm preserving. If is subjective, then is called random reflexive. In [13] Guo proved that the random reflexivity is independent of a special choice of and . The following propositions are very essential relations, which are established by Guo in [12, 21], between classical reflexive spaces and random reflexive modules.

Proposition 17 (see [21]). * is random reflexive if and only if is a reflexive Banach space.*

Proposition 18 (see [12]). *Let be an module over with base . Then is random reflexive if and only if is reflexive, where .*

Proposition 19 (see [12]). *Let be an module over with base , and a pair of Hölder conjugate numbers. Then is isometrically isomorphic with the classical conjugate space of , denoted by , under the canonical mapping defined as follows. For each , (denoting ) is defined by for all .*

#### 3. Some Basic Properties of -Valued Lower Semicontinuous Functions

In this section, we give some basic properties of -valued lower semicontinuous functions. First, we recall the definition of -valued lower semicontinuous functions under two kinds of topologies, which was presented by Guo in [14] for the first time.

Let be a left module over the algebra . The effective domain of function is denoted by . The epigraph of is denoted by . The function is called proper if on for every and .

*Definition 20 (see [14]). *Let be a left module over the algebra and .(1) is -convex if for all and in and such that (here we make the convention that and ).(2) has the local property if for all and .(3) is regular if for all and .

*Definition 21 (see [14]). *Let be an module over with base . A function is called -lower semicontinuous if is closed in . A function is called -lower semicontinuous if is closed in .

Proposition 22 (see [14]). *Let be an module over with base such that has the countable concatenation property and a function with the local property. Then the following are equivalent to each other:*(1)* is **-lower semicontinuous;*(2)* is **-closed for each **;*(3)* for each ** and each net ** in ** such that ** is **-convergent to **, where *.

*Remark 23. *Proposition 22 first occurred in [16] where the countable concatenation property of was not assumed, but this condition should be added (see [14] for details).

For -lower semicontinuous functions, we only have the following proposition.

Proposition 24 (see [14]). *Let be an module over with base and a function. Then one has the following statements:*(1)* is **-lower semicontinuous if ** for each ** and each net ** in ** such that ** is *-*convergent to *;(2)* is *-*closed for each ** if ** is *-*lower semicontinuous. *

If we define to be lower semicontinuous via “ for all net in such that it converges in the -topology to some ”, the notion is, however, meaningless in the random setting, since we can construct an module and a -continuous -convex function from to , whereas is not a -lower semicontinuous function. Hence, we cannot use this inequality for -lower semicontinuous functions. Since this inequality is very important for the proof of Theorem 1 (see Section 4 for details), we can only establish -valued minimax theorems for -lower semicontinuous functions.

Proposition 25 (see [14]). *Let be an module over with base such that has the countable concatenation property and a function with the local property. Then is -lower semicontinuous if and only if is -lower semicontinuous, specially this is true for an -convex function .*

Now, we give some important properties of -valued lower semicontinuous functions on modules. To pave the way for Theorem 26, we first introduce some notation: if is an module over with base , denotes the set of all probability measures equivalent to on , , where , and denotes the norm on , namely, for any .

Theorem 26. *Let be a random reflexive module over with base , a -closed, -convex, and a.s. bounded set with the countable concatenation property and a -lower semicontinuous function with the local property. If is proper, then is bounded from below on .*

*Proof. *Let ; then it is easy to see that . If is not bounded from below on , then there exists such that and
on . Since is -convex and has the local property, it is easy to see that is directed. Hence there exists a sequence such that . Let ; then it is clear that is -closed by Definition 21. Since has the local property and has the countable concatenation property, for any , we have that has the countable concatenation property and is -closed by Proposition 15. By the fact that is a.s. bounded, we can define a probability measure on by , where . Then is equivalent to and , for any , which means that is bounded in . Noting that replacing the probability measure of the base space with a probability measure does not change the -topology of , for any given , we can obtain that is norm-closed and convex in . Since is a random reflexive module, we have that is reflexive normed space from Proposition 18. Hence is compact under the weak topology of the normed space . Let , then one can obtain that has the finite intersection property and . Let ; then
on , which it contradicts to the fact that is proper.

For giving Theorem 28, we need to introduce the following Proposition 27, which was established by Guo and Yang in [22] for studying Ekeland's variational principle for -valued functions on modules.

Proposition 27 (see [22]). *Let be an module over with base , a subset with the countable concatenation property and have the local property. If is proper and bounded from below on (resp., bounded from above on ), then for each , there exists such that (accordingly, .*

Theorem 28. *Let be a random reflexive module over with base , a -closed, -convex and a.s. bounded set with the countable concatenation property and a -lower semicontinuous and -convex function with the local property. If is proper, then there exists such that .*

*Proof. *Let and . It is clear that and by Theorem 26. Take , for all , then is a countable partition of to . For any ; define a function as follows:
Since has the local property, for any , we have that and . Because is proper and is -lower semicontinuous and -convex, it is clear that is proper, -lower semicontinuous, and -convex. Next, we prove that has the local property. We need only to prove that
In fact, since has the local property, for any , we have that
Let for any . It is easy to see that is a bounded and convex set. Since is -convex, -closed, and a.s. bounded in and has the countable concatenation, we can obtain that is -closed in by Proposition 15. It is easy to see that is convex and -closed in from the fact that the topology induced by is stronger than the -topology. Since is random reflexive, is reflexive normed space, and is compact in under the weak topology of . For any , define . It is clear that by Proposition 27. Since is -lower semicontinuous, we have that is -closed. Thus, we have that is -convex and -closed in by the fact that has the local property and Proposition 15. By Hahn-Banach theorem, we have that is closed under the weak topology of . Take
it is easy to prove that has the finite intersection property. Since is compact under the weak topology of , we have that . Let for any and
We have that and

This completes the proof.

*Definition 29. *Let be an module over with base , and a -convex subset in . is called strictly -convex if
for any , and on .

Corollary 30. *Let be a random reflexive module over with base , a -closed, -convex and a.s. bounded set with the countable concatenation property, and a -lower semicontinuous and strictly -convex function with the local property. If is proper, then there exists an unique such that .*

#### 4. Main Results

Now, we give the definition of random saddle points.

*Definition 31. *Let and be any two nonempty sets, and . Then is called a random saddle point of with respect to if

*Remark 32. *Let and be any two nonempty sets, . It is easy to see that the following statements are equivalent:(1) is a random saddle point of with respect to ;(2);(3).

Before giving the proof of main result in this paper, we first recall the definition of random strictly convex module, which is presented by Guo and Zeng in [23] for the first time.

*Definition 33 (see [23]). *An module is said to be random strictly convex if for any and such that , then there exist and such that , on and .

*Definition 34 (see [24]). *Let be an module over with base , and . Then and are called -independent on if whenever such that .

By Definitions 33 and 34, we can obtain the following lemma easily.

Lemma 35. *Let be a random strictly convex module over with base . Then the mapping **
is strictly -convex.*

For giving the proof of Theorem 1, we need the following lemma and remark.

Lemma 36 (Mazur lemma). *Let be a normed space, converge to under the weak topology on . Then there exists a sequence such that it converges to in norm, where
*

*Remark 37. *Let converge to uniformly. Then we can obtain a net such that it converges to under the locally -convex topology of . In fact, for any let , for all . Then is a countable partition of to . Since the sequence converges to uniformly, thus for any number , there exists such that
for any . Let
then it is easy to see that . Set ; then is directed with respect to , and one can easy to see that the net converges to under the locally -convex topology of .

With the above preparations, we now give the proof of Theorem 1.

*Proof of Theorem 1. *First, let us assume that for any , is strictly -convex on . Set and . We show that the functional has the local property and is -convex and -lower semicontinuous on . By conditions (1) and (2), it is clear that has the local property. For any , we have that
Thus, is -convex on . For any , let . Since has the local property, we have that has the countable concatenation property. Let converge to under the locally -convex topology of . By , we can obtain that
Since is -lower semicontinuous, it is easy to see that
Hence, and . So we have that is -lower semicontinuous. Similarly, replacing with , we see that has the local property and is -convex and -lower semicontinuous on .

By Theorems 26 and 28, we have that , and there exists such that
Since for all , is strictly -convex on , according to Corollary 30 there exists an unique such that
for any . Let , we need only to prove that

For every , let and , for all . It is clear that
By condition (2), we have that
namely, . Let and , for all , according to the condition that is a.s. bounded in , and it is easy to see that is a countable partition of to .

For any fixed , we have that , for all . It is easy to see that is a bounded and closed subset of . Thus, there exists a subsequence and such that converges to under the weak topology of . Without loss of generality, denote this subsequence by . By Lemma 36, there exists a sequence such that it converges to in norm, where
. Since , we have that converges in probability to . By Egoroff theorem, for any number , there exists such that and converges uniformly to 0 on . Then there is a net
as in Remark 37, which converges 0 on under the locally -convex topology of . By the construction of as in Remark 37, we have that
Since is -convex for any , we have that
Hence, one can obtain that . Because is an arbitrary nonnegative number and has the local property, we have that
for any .

Now, we prove that for any . By the definition of , it is clear that
By condition (2), we have that
Hence, we can obtainthat
Since , we can obtain that and . According to
where , it is obvious that
Since converges in probability to 0, by Egoroff theorem, we can obtain that for any , there exists such that and converges to 0 uniformly on . By Remark 37, we can construct a net as in Remark 37 such that under the locally -convex topology of . Hence, by (2) we can obtain that
Since is an arbitrary nonnegative number, it is clear that
Therefore, for any , and , for all ; namely, is a random saddle point of with respect to .

If there is such that is not strictly -convex on , define
Since is random strictly convex module, we can obtain that is strictly -convex from Lemma 35. By the similar method, we have that for any , there exists such that it is a saddle point of with respect to .

Let and be any representation element of , for all