A Minimax Theorem for 𝐿⁰-Valued Functions on Random Normed Modules

Abstract

We generalize the well-known minimax theorems to 𝐿⁰-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 𝐿⁰-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 𝐿∶𝐴×𝐵→𝑅 show that, under some suitable conditions of compactness, convexity, and continuity, the equality inf_𝑦∈𝐵sup_𝑥∈𝐴𝑓𝑥,𝑦=sup_𝑥∈𝐴inf_𝑦∈𝐵𝑓𝑥,𝑦 (1) 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 𝐿⁰-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 as RN 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 𝐿⁰-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); 𝐿⁰(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  𝒯_𝑐-closed, 𝐿⁰(ℱ)-convex subset with the countable concatenation property of 𝐸 and 𝐿∶𝐸×𝐸→𝐿⁰(ℱ). If 𝐿 satisfies the following:(1)for any fixed 𝑝∈𝐵, 𝐿(⋅,𝑝)∶𝐸→𝐿⁰(ℱ) is proper 𝐿⁰(ℱ)-convex, 𝒯_𝑐-lower semicontinuous function on 𝐴 and has the local property;(2)for any fixed 𝑢∈𝐴, −𝐿(𝑢,⋅)∶𝐸→𝐿⁰(ℱ) is proper 𝐿⁰(ℱ)-convex, 𝒯_𝑐-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, _{𝑢∈𝐴𝑝∈𝐵}𝐿𝑢,𝑝=𝐿𝑢₀,𝑝₀=_{𝑝∈𝐵𝑢∈𝐴}𝐿𝑢,𝑝. (2)

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 (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 𝒯_𝑐-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 Section 2 we will briefly collect some necessary known facts; in Section 3 we will give some basic properties of an 𝐿⁰-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 𝐿⁰(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 𝐿⁰(ℱ) has a supremum, denoted by ⋁𝐺, and an infimum, denoted by ⋀𝐺. Finally 𝐿⁰(ℱ), as a sublattice of 𝐿⁰(ℱ), is also a complete lattice in the sense that every subset with upper bound has a supremum. The pleasant properties of 𝐿⁰(ℱ) are summarized as follows.

Proposition 2 (see [19]). For every subset 𝐺 of 𝐿⁰(ℱ), there exist countable subsets {𝑎_𝑛∣𝑛∈𝑁} and {𝑏_𝑛∣𝑛∈𝑁} of 𝐺 such that ⋁𝐺=⋁_𝑛≥1𝑎_𝑛 and ⋀𝐺=⋀_𝑛≥1𝑏_𝑛. Further, if 𝐺 is directed (dually directed) with respect to ≤, then the above {𝑎_𝑛∣𝑛∈𝑁}(accordingly, {𝑏_𝑛∣𝑛∈𝑁}) can be chosen as nondecreasing (correspondingly, nonincreasing) with respect to ≤.

Specially, 𝐿⁰₊={𝜉∈𝐿⁰(ℱ)∣𝜉≥0}, 𝐿⁰₊₊={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, 𝐴^𝑐 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, 𝐼_𝐴 is the characteristic function of 𝐴, and 𝐼_𝐴̃ is used to denote the equivalence class of 𝐼_𝐴; 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_𝐴̃; one can also understand the implication of such notation as 𝐼_{[𝜉≤𝜂]}, 𝐼_[𝜉<𝜂] and 𝐼_[𝜉=𝜂].

For an arbitrarily chosen representative 𝜉⁰ 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 𝐿⁰₊(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. Let 𝐿⁰(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 of RN modules 𝐿^𝑝_ℱ(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)→𝐿⁰₊(ℱ) by |||𝑥|||_𝑝=𝐸|𝑥|^𝑝∣ℱ^1/𝑝,when⋀1≤𝑝<∞,𝜉∈𝐿⁰₊(ℱ)∣|𝑥|≤𝜉,when𝑝=+∞, (3) for all 𝑥∈𝐿⁰(ℰ).
Denote 𝐿^𝑝_ℱ(ℰ)={𝑥∈𝐿⁰(ℰ)|||𝑥|||_𝑝∈𝐿⁰₊(ℱ)}, then (𝐿^𝑝_ℱ(ℰ),|||⋅|||_𝑝) is an RN module over 𝑅 with base (Ω,ℱ,𝑃) and 𝐿^𝑝_ℱ(ℰ)=𝐿⁰(ℱ)⋅𝐿^𝑝(ℰ)={𝜉𝑥∣𝜉∈𝐿⁰(ℱ) and 𝑥∈𝐿^𝑝(ℰ)}.

To put some important classes of stochastic processes into the framework of RN modules, Guo constructed a more general RN module 𝐿^𝑝_ℱ(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→𝐿⁰₊(ℱ) by |||⋅|||_𝑝=𝐸‖𝑥‖^𝑝∣ℱ^1/𝑝,when⋀1≤𝑝<∞,𝜉∈𝐿⁰₊(ℱ)∣‖𝑥‖≤𝜉,when𝑝=+∞, (4) for all 𝑥∈𝐸.
Denote 𝐿^𝑝_ℱ(𝐸)={𝑥∈𝑆∣|||𝑥|||_𝑝∈𝐿⁰₊(ℱ)}; then (𝐿^𝑝_ℱ(𝐸),|||⋅|||_𝑝) is an RN module over 𝐾 with base (Ω,ℱ,𝑃). When 𝐸=𝐿⁰(ℰ), 𝐿^𝑝_ℱ(𝐸) is exactly 𝐿^𝑝_ℱ(ℰ).

Remark 7. For a given RN 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 𝑔_𝑝=∫_Ω𝑔^𝑝1/𝑝,if1≤𝑝<+∞,the𝑃-essentialsupremum,if𝑝=+∞. (5) Let 𝐿^𝑝(𝐸)={𝑔∈𝐸∶‖𝑔‖_𝑝<+∞}. As mentioned in [13], (𝐿^𝑝(𝐸),‖⋅‖_𝑝) is a normed space over 𝐾 and is further a Banach space if (𝐸,‖⋅‖) is complete.

For each RN 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 𝑁_𝜃(𝜀,𝜆)={𝑥∈𝐸∣𝑃{𝜔∈Ω∣‖𝑥‖(𝜔)<𝜀}>1−𝜆}; then {𝑁_𝜃(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 𝒯_𝜀,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 for 𝐿⁰(ℱ,𝐾) is exactly the topology of convergence in probability 𝑃, and (𝐿⁰(ℱ,𝐾),𝒯_𝜀,𝜆) is a topological algebra over 𝐾.(2)If (𝐸,‖⋅‖) is an 𝑅𝑁 modules, then (𝐸,𝒯_𝜀,𝜆) is a topological module over the topological algebra𝐿⁰(𝐹,𝐾).(3)A net {𝑥_𝛿,𝛿∈Γ} converges in the (𝜀,𝜆)-topology to some 𝑥 in 𝐸 if and only if {‖𝑥_𝛿−𝑥‖,𝛿∈Γ} converges in probability 𝑃 to 0.

The following locally 𝐿⁰-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∈𝐿⁰₊₊, let 𝑁_𝜃(𝜀)={𝑥∈𝐸∣‖𝑥‖≤𝜀}. A subset 𝐺 of 𝐸 is called 𝒯_𝑐-open if for each 𝑥∈𝐺 there exists some 𝑁_𝜃(𝜀) such that 𝑥+𝑁_𝜃(𝜀)⊂𝐺, and 𝒯_𝑐 denotes the family of 𝒯_𝑐-open subsets of 𝐸. Then it is easy to see that (𝐸,𝒯_𝑐) is a Hausdorff topological group with respect to the addition on 𝐸. 𝒯_𝑐 is called the locally 𝐿⁰-convex topology for 𝐸 induced by ‖⋅‖.

From now on, the locally 𝐿⁰-convex topology for each random locally convex space is always denoted by 𝒯_𝑐 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 locally 𝐿⁰-convex topology;(2)𝐸 is a topological module over the topological ring 𝐿⁰(ℱ,𝐾) when 𝐸 and 𝐿⁰(ℱ,𝐾) are endowed with their respective locally 𝐿⁰-convex topologies;(3)a net {𝑥_𝛼∣𝛼∈Γ} in 𝐸 converges in the locally 𝐿⁰-convex topology to 𝑥∈𝐸 if and only if {‖𝑥_𝛼−𝑥‖∣𝛼∈Γ} converges in the locally 𝐿⁰-convex topology of 𝐿⁰(ℱ,𝐾) to 0.

𝒯_𝑐 is called locally 𝐿⁰-convex because it has a striking local base 𝒰_𝜃={𝐵(𝜀)∣𝜀∈𝐿⁰₊₊}, each member 𝑈 of which is as follows:(i)𝐿⁰-convex: 𝜉⋅𝑥+(1−𝜉)⋅𝑦∈𝑈 for any 𝑥,𝑦∈𝑈 and 𝜉∈𝐿⁰₊ such that 0⩽𝜉⩽1;(ii)𝐿⁰-absorbent: there is 𝜉∈𝐿⁰₊₊ 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 𝒯_𝑐. 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,𝒯_𝑐) is not a topological linear space since the map 𝐾→(𝐸,𝒯_𝑐), 𝛼→𝛼⋅𝑥, is not necessarily continuous for 𝑥≠𝜃; see [16] for details.

In the sequel of this paper, for a subset 𝐺 of an RN module (𝐸,‖⋅‖), 𝐺_𝜀,𝜆 denotes the 𝒯_𝜀,𝜆-closure of 𝐺, and 𝐺_𝑐 denotes the 𝒯_𝑐-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]). Let 𝐸 be a left module over the algebra 𝐿⁰(ℱ,𝐾). A formal sum Σ_𝑛≥1𝐼̃_𝐴𝑛𝑥_𝑛 for some countable partition {𝐴_𝑛,𝑛∈𝑁} of Ω to ℱ and some sequence {𝑥_𝑛∣𝑛∈𝑁} in 𝐸 is called a countable concatenation of {𝑥_𝑛∣𝑛∈𝑁} with respect to {𝐴_𝑛,𝑛∈𝑁}. Furthermore a countable concatenation Σ_𝑛≥1𝐼̃_𝐴𝑛𝑥_𝑛 is well defined or Σ_𝑛≥1𝐼̃_𝐴𝑛𝑥_𝑛∈𝐸 if there is 𝑥∈𝐸 such that 𝐼̃_𝐴𝑛𝑥=𝐼̃_𝐴𝑛𝑥_𝑛, for all 𝑛∈𝑁. A subset 𝐺 of 𝐸 is said to have the countable concatenation property if every countable concatenation Σ_𝑛≥1𝐼̃_𝐴𝑛𝑥_𝑛 with 𝑥_𝑛∈𝐺 for each 𝑛∈𝑁 still belongs to 𝐺; namely, Σ_𝑛≥1𝐼̃_𝐴𝑛𝑥_𝑛 is well defined and there exists 𝑥∈𝐺 such that 𝑥=Σ_𝑛≥1𝐼̃_𝐴𝑛𝑥_𝑛.

Proposition 14 (see [13]). Let (𝐸,‖⋅‖) be an 𝑅𝑁 module over 𝐾 with base (Ω,ℱ,𝑃). Then 𝐸 is 𝒯_𝜀,𝜆-complete if and only if E is 𝒯_𝑐-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 of RN 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 𝐿⁰₊(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^∗→𝐿⁰₊(ℱ,𝑅) by ‖𝑓‖^∗=⋀{𝜉∈𝐿⁰₊(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), 𝐸^∗_𝜀,𝜆 the 𝐿⁰(ℱ,𝐾)-module of continuous module homomorphisms from (𝐸,𝒯_𝜀,𝜆) to (𝐿⁰(ℱ,𝐾),𝒯_𝜀,𝜆), and 𝐸^∗_𝑐 the 𝐿⁰(𝐹,𝐾)-module of continuous module homomorphisms from (𝐸,𝒯_𝑐) to (𝐿⁰(𝐹,𝐾),𝒯_𝑐), then it was proved that 𝐸^∗_𝜀,𝜆=𝐸^∗_𝑐. 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 𝒯_𝜀,𝜆 and 𝒯_𝑐. 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]). 𝐿⁰(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 (𝐿^𝑝(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 (𝐿^𝑞(𝐸^∗),‖⋅‖_𝑞) is isometrically isomorphic with the classical conjugate space of (𝐿^𝑝(𝐸),‖⋅‖), denoted by (𝐿^𝑝(𝐸))^′, under the canonical mapping 𝑇∶𝐿^𝑞(𝐸^∗)→(𝐿^𝑝(𝐸))^′ defined as follows. For each 𝑓∈𝐿^𝑞(𝐸^∗), 𝑇_𝑓 (denoting 𝑇(𝑓))∶𝐿^𝑝(𝐸)→𝐾 is defined by 𝑇_𝑓(𝑔)=∫_Ω𝑓(𝑔)𝑑𝑃 for all 𝑔∈𝐿^𝑝(𝐸).

3. Some Basic Properties of 𝐿⁰-Valued Lower Semicontinuous Functions

In this section, we give some basic properties of 𝐿⁰-valued lower semicontinuous functions. First, we recall the definition of 𝐿⁰-valued lower semicontinuous functions under two kinds of topologies, which was presented by Guo in [14] for the first time.

Let 𝐸 be a left module over the algebra 𝐿⁰(ℱ). The effective domain of function 𝑓∶𝐸→𝐿⁰(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]). Let 𝐸 be a left module over the algebra 𝐿⁰(ℱ) and 𝑓∶𝐸→𝐿⁰(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∈𝐿⁰₊ such that 0≤𝜉≤1 (here we make the convention that 0⋅(±∞)=0 and ∞−∞=∞).(2)𝑓 has the local property if 𝐼_𝐴̃𝑓(𝑥)=𝐼_𝐴̃𝑓(𝐼_𝐴̃𝑥) for all 𝑥∈𝐸 and 𝐴∈ℱ.(3)𝑓 is regular if 𝐼_𝐴̃𝑓(𝑥)=𝑓(𝐼_𝐴̃𝑥) 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→𝐿⁰(ℱ) is called 𝒯_𝑐-lower semicontinuous if epi(𝑓) is closed in (𝐸,𝒯_𝑐)×(𝐿⁰(ℱ),𝒯_𝑐). A function 𝑓∶𝐸→𝐿⁰(ℱ) is called 𝒯_𝜀,𝜆-lower semicontinuous if epi(𝑓) is closed in (𝐸,𝒯_𝜀,𝜆)×(𝐿⁰(ℱ),𝒯_𝜀,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→𝐿⁰(ℱ) a function with the local property. Then the following are equivalent to each other:(1)𝑓 is 𝒯_𝑐-lower semicontinuous;(2){𝑥∈𝐸∣𝑓(𝑥)≤𝑟} is 𝒯_𝑐-closed for each 𝑟∈𝐿⁰(ℱ);(3)lim_𝛼𝑓(𝑥_𝛼)≥𝑓(𝑥₀) for each 𝑥₀∈𝐸 and each net {𝑥_𝛼,𝛼∈Γ} in 𝐸 such that {𝑥_𝛼,𝛼∈Γ} is 𝒯_𝑐-convergent to 𝑥₀, where lim_𝛼𝑓(𝑥_𝛼)=⋁_𝛼∈Γ(⋀_𝛽≥𝛼𝑓(𝑥_𝛽)).

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

For 𝒯_𝜀,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→𝐿⁰(ℱ) a function. Then one has the following statements:(1)𝑓 is 𝒯_𝜀,𝜆-lower semicontinuous if lim_𝛼𝑓(𝑥_𝛼)≥𝑓(𝑥₀) for each 𝑥₀∈𝐸 and each net {𝑥_𝛼,𝑥∈Γ} in 𝐸 such that {𝑥_𝛼,𝛼∈Γ} is 𝒯_𝜀,𝜆-convergent to 𝑥₀;(2){𝑥∈𝐸∣𝑓(𝑥)≤𝑟} is 𝒯_𝜀,𝜆-closed for each 𝑟∈𝐿⁰(ℱ) if 𝑓 is 𝒯_𝜀,𝜆-lower semicontinuous.

If we define 𝑓 to be lower semicontinuous via “lim_𝛼𝑓(𝑥_𝛼)≥𝑓(𝑥) for all net {𝑥_𝛼,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 𝒯_𝜀,𝜆-continuous 𝐿⁰-convex function 𝑓 from 𝐸 to 𝐿⁰(ℱ), whereas 𝑓 is not a 𝒯_𝜀,𝜆-lower semicontinuous function. Hence, we cannot use this inequality for 𝒯_𝜀,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 𝐿⁰-valued minimax theorems for 𝒯_𝑐-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→𝐿⁰(ℱ) a function with the local property. Then 𝑓 is 𝒯_𝜀,𝜆-lower semicontinuous if and only if 𝑓 is 𝒯_𝑐-lower semicontinuous, specially this is true for an 𝐿⁰(ℱ)-convex function 𝑓.

Now, we give some important properties of 𝐿⁰-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), 𝐿^𝑝_𝑄(𝐸)={𝑥∈𝐸∣∫_Ω‖𝑥‖^𝑝𝑑𝑄<+∞}, where 𝑄∈Π, and ‖⋅‖^𝑄_𝑝 denotes the norm on 𝐿^𝑝_𝑄(𝐸), namely, ‖𝑥‖^𝑄_𝑝=(∫_Ω‖𝑥‖^𝑝𝑑𝑄)^1/𝑝 for any 𝑥∈𝐿^𝑝_𝑄(𝐸).

Theorem 26. Let (𝐸,‖⋅‖) be a random reflexive 𝑅𝑁 module over 𝑅 with base (Ω,ℱ,𝑃), 𝐺⊂𝐸 a 𝒯_𝑐-closed, 𝐿⁰(ℱ)-convex, and a.s. bounded set with the countable concatenation property and 𝑓∶𝐸→𝐿⁰(ℱ) a 𝒯_𝑐-lower semicontinuous function with the local property. If 𝑓∣_𝐺 is proper, then 𝑓 is bounded from below on 𝐺.

Proof. Let 𝜂=∨{‖𝑥‖∶𝑥∈𝐺}; then it is easy to see that 𝜂∈𝐿⁰₊. If 𝑓 is not bounded from below on 𝐺, then there exists 𝐵∈ℱ such that 𝑃(𝐵)>0 and 𝑓(𝑥)∣𝑥∈𝐺=−∞ (6) 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 {𝑥_𝑛,𝑛∈𝑁} such that {𝑓(𝑥_𝑛),𝑛∈𝑁}↘⋀{𝑓(𝑥)∣𝑥∈𝐺}. Let 𝐺_𝑛={𝑥∈𝐸∣𝑓(𝑥)≤𝑓(𝑥_𝑛)}⋂𝐺; then it is clear that 𝐺_𝑛 is 𝒯_𝑐-closed by Definition 21. Since 𝑓 has the local property and 𝐺 has the countable concatenation property, for any 𝑛∈𝑁, we have that 𝐺_𝑛 has the countable concatenation property and 𝐺_𝑛 is 𝒯_𝜀,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 (𝐿²_𝑄(𝐸),‖⋅‖^𝑄₂). 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 𝐺_𝑛 is norm-closed and convex in (𝐿²_𝑄(𝐸),‖⋅‖^𝑄₂). Since (𝐸,‖⋅‖) is a random reflexive RN module, we have that (𝐿²_𝑄(𝐸),‖⋅‖^𝑄₂) is reflexive normed space from Proposition 18. Hence 𝐺_𝑛 is compact under the weak topology of the normed space 𝐿²_𝑄(𝐸). Let 𝒪={𝐺_𝑛∣𝑛∈𝑁}, then one can obtain that 𝒪 has the finite intersection property and ⋂𝒪≠∅. Let 𝑥^∗∈⋂𝒪; then 𝑓𝑥^∗=−∞ (7) on 𝐵, which it contradicts to the fact that 𝑓|_𝐺 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 𝐿⁰-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→𝐿⁰(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∈𝐿⁰₊₊(ℱ), there exists 𝑥_𝜀∈𝐺 such that 𝑓(𝑥_𝜀)≤⋀𝑓(𝐺)+𝜀 (accordingly, 𝑓(𝑥_𝜀)≥⋁𝑓(𝐺)−𝜀).

Theorem 28. Let (𝐸,‖⋅‖) be a random reflexive 𝑅𝑁 module over 𝑅 with base (Ω,ℱ,𝑃), 𝐺⊂𝐸 a 𝒯_𝑐-closed, 𝐿⁰(ℱ)-convex and a.s. bounded set with the countable concatenation property and 𝑓∶𝐸→𝐿⁰(ℱ) a 𝒯_𝑐-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. Let 𝜉=⋁{‖𝑥‖∣𝑥∈𝐺} and 𝜂=⋀𝑓(𝐺). It is clear that 𝜉∈𝐿⁰₊ and 𝜂∈𝐿⁰(ℱ) by Theorem 26. Take 𝐵_𝑗=[𝑗−1≤𝜉<𝑗], for all 𝑗∈𝑁, then {𝐵_𝑗∣𝑗=1,2,…} is a countable partition of Ω to ℱ. For any 𝑗∈𝑁; define a function 𝑓_𝑗∶𝐼̃_𝐵𝑗⋅𝐸→𝐿⁰(ℱ) as follows: 𝑓_𝑗𝐼̃_𝐵𝑗𝑥=𝐼̃_𝐵𝑗𝑓(𝑥),∀𝑥∈𝐸. (8) Since 𝑓 has the local property, for any 𝑗∈𝑁, we have that 𝑓_𝑗(𝐼̃_𝐵𝑗𝑥)=𝐼̃_𝐵𝑗𝑓(𝐼̃_𝐵𝑗𝑥) and 𝐼̃_𝐵𝑗⋅⋀𝑓(𝐺)=⋀𝑓_𝑗(𝐼̃_𝐵𝑗⋅𝐺). Because 𝑓|_𝐺 is proper and 𝑓 is 𝒯_𝑐-lower semicontinuous and 𝐿⁰(ℱ)-convex, it is clear that 𝑓_𝑗|̃𝐼_{𝐵𝑗⋅𝐺} is proper, 𝒯_𝑐-lower semicontinuous, and 𝐿⁰(ℱ)-convex. Next, we prove that 𝑓_𝑗 has the local property. We need only to prove that 𝐼_𝐴̃𝑓_𝑗𝐼_𝐴̃𝐼̃_𝐵𝑗𝑥=𝐼_𝐴̃𝑓_𝑗𝐼̃_𝐵𝑗𝑥,∀𝐴∈ℱ𝐵_𝑗. (9) In fact, since 𝑓 has the local property, for any 𝐴∈ℱ⋂𝐵_𝑗, we have that 𝐼_𝐴̃𝑓_𝑗𝐼̃_𝐵𝑗𝑥=𝐼_𝐴̃𝐼̃_𝐵𝑗𝑓(𝑥)=𝐼_𝐴̃𝐼̃_𝐵𝑗𝑓𝐼̃_𝐵𝑗𝑥=𝐼_𝐴̃𝐼̃_𝐵𝑗𝑓𝐼_𝐴̃𝐼̃_𝐵𝑗𝑥=𝐼_𝐴̃𝑓_𝑗𝐼_𝐴̃𝐼̃_𝐵𝑗𝑥. (10) Let 𝜂_𝑗=𝐼̃_𝐵𝑗⋅𝜂 for any 𝑗∈𝑁. It is easy to see that 𝐼_𝑗̃𝐺⊆(𝐿²(𝐸),‖⋅‖₂) is a bounded and convex set. Since 𝐼̃_𝐵𝑗𝐺 is 𝐿⁰(ℱ)-convex, 𝒯_𝑐-closed, and a.s. bounded in 𝐸 and has the countable concatenation, we can obtain that 𝐼̃_𝐵𝑗𝐺 is 𝒯_𝜀,𝜆-closed in 𝐸 by Proposition 15. It is easy to see that 𝐼̃_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 𝐼̃_𝐵𝑗𝐺 is compact in 𝐿²(𝐸) under the weak topology of 𝐿²(𝐸). For any 𝜀∈𝐿⁰₊₊, define 𝐺_𝑗(𝜀)={𝐼̃_𝐵𝑗𝑥∣𝑓_𝑗(𝐼̃_𝐵𝑗𝑥)≤𝐼̃_𝐵𝑗𝜂+𝜀,𝑥∈𝐺}. It is clear that 𝐺_𝑗(𝜀)≠∅ by Proposition 27. Since 𝑓_𝑗|̃𝐼_{𝐵𝑗⋅𝐺} is 𝒯_𝑐-lower semicontinuous, we have that 𝐺_𝑗(𝜀) is 𝒯_𝑐-closed. Thus, we have that 𝐺_𝑗(𝜀) is 𝐿⁰(ℱ)-convex and ‖⋅‖₂-closed in 𝐿²(𝐸) by the fact that 𝑓_𝑗 has the local property and Proposition 15. By Hahn-Banach theorem, we have that 𝐺_𝑗(𝜀) is closed under the weak topology of 𝐿²(𝐸). Take 𝒪=𝐺_𝑗(𝜀)∣𝜀∈𝐿⁰₊₊, (11) it is easy to prove that 𝒪 has the finite intersection property. Since 𝐼̃_𝐵𝑗𝐺 is compact under the weak topology of 𝐿²(𝐸), we have that ⋂𝒪≠∅. Let 𝑥_𝑗∈⋂𝒪 for any 𝑗∈𝑁 and 𝑥^∗=^∞_𝑗=1𝐼̃_𝐵𝑗⋅𝑥_𝑗. (12) We have that 𝑓_𝑗(𝑥_𝑗)=𝜂_𝑗 and 𝑓𝑥^∗=𝑓(𝐺). (13)
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→𝐿⁰(ℱ) is called strictly 𝐿⁰(ℱ)-convex if 𝑓𝛼𝑥+(1−𝛼)𝑦<𝛼𝑓(𝑥)+(1−𝛼)𝑓𝑦, (14) 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 𝒯_𝑐-closed, 𝐿⁰(ℱ)-convex and a.s. bounded set with the countable concatenation property, and 𝑓∶𝐸→𝐿⁰(ℱ) a 𝒯_𝑐-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. Let 𝐴 and 𝐵 be any two nonempty sets, (𝑢₀,𝑝₀)∈𝐴×𝐵 and 𝐿∶𝐴×𝐵→𝐿⁰(ℱ). Then (𝑢₀,𝑝₀) is called a random saddle point of 𝑓 with respect to 𝐴×𝐵 if 𝐿𝑢₀,𝑝≤𝐿𝑢₀,𝑝₀≤𝐿𝑢,𝑝₀,∀𝑢∈𝐴,𝑝∈𝐵. (15)

Remark 32. Let 𝐴 and 𝐵 be any two nonempty sets, 𝐿∶𝐴×𝐵→𝐿⁰(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 convex RN module, which is presented by Guo and Zeng in [23] for the first time.

Definition 33 (see [23]). An RN 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∈𝐿⁰₊ such that 𝑃(𝐴)>0, 𝜉>0 on 𝐴 and 𝐼_𝐴̃𝑥=𝜉(𝐼_𝐴̃𝑦).

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_𝐹̃=𝜂𝐼_𝐹̃=0 whenever 𝜉,𝜂∈𝐿⁰(ℱ) such that 𝜉𝐼_𝐹̃𝑥+𝜂𝐼_𝐹̃𝑦=𝜃.

By Definitions 33 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)𝑓(𝑥)=‖𝑥‖² (16) is strictly 𝐿⁰(ℱ)-convex.

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

Lemma 36 (Mazur lemma). Let (𝑋,‖⋅‖) be a normed space, {𝑢_𝑛∈𝑋,𝑛∈𝑁} converge to 𝑢 under the weak topology on 𝑋. Then there exists a sequence {𝑣_𝑛∈𝑋,𝑛∈𝑁} such that it converges to 𝑢 in norm, where 𝑣_𝑛=^𝑁𝑛_𝑘=𝑛𝜆_𝑘𝑢_𝑘,_{𝑁𝑛𝑘=𝑛}𝜆_𝑘=1,𝜆_𝑘≥0. (17)

Remark 37. Let {𝜉_𝑛∈𝐿⁰(ℱ),𝑛∈𝑁} converge to 𝜉 uniformly. Then we can obtain a net {𝜉_𝜀∈𝐿⁰(ℱ)∣𝜀∈𝐿⁰₊₊,𝜀≤1} such that it converges to 𝜉 under the locally 𝐿⁰-convex topology of (𝐿⁰(ℱ,𝑅),|⋅|). In fact, for any 𝜀∈𝐿⁰₊₊ let 𝐴₁=[𝜀>1],𝐴_𝑖=[1/(𝑖+1)<𝜀≤1/𝑖], for all 𝑖∈𝑁. Then {𝐴_𝑛,𝑛∈𝑁} is a countable partition of Ω to ℱ. Since the sequence {𝜉_𝑛,𝑛∈𝑁} converges to 𝜉 uniformly, thus for any number 𝑘>0, there exists 𝑁(𝑘)∈𝑁 such that 𝜉_𝑛<1−𝜉𝑘 (18) for any 𝑛>𝑁(𝑘). Let 𝜉_𝜀=^∞_𝑖=1𝐼̃_𝐴𝑖𝜉_𝑁(𝑖)+1; (19) then it is easy to see that |𝜉_𝜀−𝜉|<𝜀. Set ∧={𝜀∈𝐿⁰₊₊∣𝜀≤1}; then ∧ is directed with respect to ≤, and one can easy to see that the net {𝜉_𝜀,𝜀∈∧} converges to 𝜉 under the locally 𝐿⁰-convex topology of (𝐿⁰(ℱ),|⋅|).

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

Proof of Theorem 1. First, let us assume that for any 𝑝∈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 𝒯_𝑐-lower semicontinuous on 𝐴. By conditions (1) and (2), it is clear that 𝐹 has the local property. For any 𝑥₁,𝑥₂∈𝐴,𝛼∈𝐿⁰₊, we have that 𝐹𝛼𝑥₁+(1−𝛼)𝑥₂=_𝑝∈𝐵𝐿𝛼𝑥₁+(1−𝛼)𝑥₂≤,𝑝_𝑝∈𝐵𝛼𝐿𝑥₁,𝑝+(1−𝛼)𝐿𝑥₂,𝑝≤𝛼_𝑝∈𝐵𝐿𝑥₁,𝑝+(1−𝛼)_𝑝∈𝐵𝐿𝑥₂𝑥,𝑝=𝛼𝐹₁+(1−𝛼)𝐹𝑥₂. (20) Thus, 𝐹 is 𝐿⁰(ℱ)-convex on 𝐴. For any 𝑟∈𝐿⁰(ℱ,𝑅), let 𝐴_𝑟={𝑢∈𝐴∣𝐹(𝑢)≤𝑟}. Since 𝐹 has the local property, we have that 𝐴_𝑟 has the countable concatenation property. Let {𝑥_𝛼,𝛼∈∧}⊂𝐴_𝑟 converge to 𝑥₀ under the locally 𝐿⁰-convex topology of 𝐸. By 𝐹(𝑥_𝛼)≤𝑟, we can obtain that 𝐿𝑥_𝛼,𝑝≤𝑟,∀𝑝∈𝐵. (21) Since 𝐿(⋅,𝑝) is 𝒯_𝑐-lower semicontinuous, it is easy to see that 𝐿𝑥₀,𝑝≤𝑟,∀𝑝∈𝐵. (22) Hence, 𝐹(𝑥₀)≤𝑟 and 𝑥₀∈𝐴_𝑟. So we have that 𝐹 is 𝒯_𝑐-lower semicontinuous. Similarly, replacing 𝐿 with −𝐿, we see that −𝐺 has the local property and −𝐺 is 𝐿⁰(ℱ)-convex and 𝒯_𝑐-lower semicontinuous on 𝐵.
By Theorems 26 and 28, we have that ⋁_𝑝∈𝐵𝐺(𝑝)∈𝐿⁰(ℱ), and there exists 𝑝₀∈𝐵 such that 𝐺𝑝₀=_𝑝∈𝐵𝐺𝑝. (23) Since for all 𝑝∈𝐵, 𝐿(⋅,𝑝) is strictly 𝐿⁰(ℱ)-convex on 𝐴, according to Corollary 30 there exists an unique 𝑢_𝑝∈𝐴 such that 𝐺𝑝=_𝑢∈𝐴𝐿𝑢,𝑝=𝐿𝑢_𝑝,𝑝, (24) for any 𝑝∈𝐵. Let 𝑢₀=𝑢_𝑝0, we need only to prove that 𝐺𝑝₀≥𝐿𝑢₀,𝑝,∀𝑝∈𝐵. (25)
For every 𝑝∈𝐵, let 𝑝_𝑛=(1−𝑛⁻¹)𝑝₀+𝑛⁻¹𝑝 and 𝑢_𝑛=𝑢_𝑝𝑛, for all 𝑛∈𝑁. It is clear that 𝐺𝑝₀≥𝐺𝑝_𝑛=𝐿𝑢_𝑛,𝑝_𝑛. (26) By condition (2), we have that 𝐺𝑝₀≥𝐿𝑢_𝑛,𝑝_𝑛≥1−𝑛⁻¹𝐿𝑢_𝑛,𝑝₀+𝑛⁻¹𝐿𝑢_𝑛,𝐺𝑝,𝑝₀≥1−𝑛⁻¹𝐺𝑝₀+𝑛⁻¹𝐿𝑢_𝑛,𝑝, (27) namely, 𝐺(𝑝₀)≥𝐿(𝑢_𝑛,𝑝). Let 𝜂=⋁{‖𝑥‖∣𝑥∈𝐴} and 𝐶_𝑛={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 {𝐶_𝑛,𝑛∈𝑁} is a countable partition of Ω to ℱ.
For any fixed 𝑖∈𝑁, we have that 𝐼̃_𝐶𝑖⋅𝑢_𝑛∈𝐿²(𝐸), for all 𝑛∈𝑁. It is easy to see that 𝐼̃_𝐶𝑖⋅𝐴 is a bounded and closed subset of (𝐿²(𝐸),‖⋅‖₂). Thus, there exists a subsequence {𝐼̃_𝐶𝑖⋅𝑢_𝑛𝑘,𝑘∈𝑁} and 𝜔_𝑖∈𝐼̃_𝐶𝑖⋅𝐴 such that {𝐼̃_𝐶𝑖⋅𝑢_𝑛𝑘,𝑘∈𝑁} converges to 𝜔_𝑖 under the weak topology of 𝐿²(𝐸). Without loss of generality, denote this subsequence by {𝐼̃_𝐶𝑖⋅𝑢_𝑛,𝑛∈𝑁}. By Lemma 36, there exists a sequence {𝑣_𝑛∈𝐼̃_𝐶𝑖⋅𝐴,𝑛∈𝑁} such that it converges to 𝜔_𝑖 in norm, where 𝑣_𝑛=^𝑁𝑛_𝑘=𝑛𝜆_𝑘𝑢_𝑘, (28)∑^𝑁𝑛_𝑘=𝑛𝜆_𝑘=1,𝜆_𝑘≥0. Since ‖𝑣_𝑛−𝜔_𝑖‖₂→0, we have that {‖𝑣_𝑛−𝜔_𝑖‖,𝑛∈𝑁} converges in probability 𝑃 to 𝜃. By Egoroff theorem, for any number 𝛿>0, there exists 𝐶_𝛿∈ℱ such that 𝑃(𝐶_𝛿)<𝛿 and {‖𝑣_𝑛−𝜔_𝑖‖,𝑛∈𝑁} converges uniformly to 0 on 𝐶^𝑐_𝛿. Then there is a net 𝑣_𝜀−𝜔_𝑖,𝜀∈∧ (29) as in Remark 37, which converges 0 on 𝐶^𝑐_𝛿 under the locally 𝐿⁰-convex topology of (𝐿⁰(ℱ),|⋅|). By the construction of {‖𝑣_𝜀−𝜔_𝑖‖,𝜀∈∧} as in Remark 37, we have that lim_𝑛→∞𝐿𝑣_𝑛,𝑝≥lim_𝜀∈∧𝐿𝑣_𝜀,𝑝. (30) Since 𝐿(⋅,𝑝) is 𝐿⁰(ℱ)-convex for any 𝑝∈𝐵, we have that 𝐼̃_𝐶𝑐𝛿𝐼̃_𝐶𝑖𝐺𝑝₀≥𝐼̃_𝐶𝑐𝛿𝐼̃_𝐶𝑖lim^𝑁_𝑛→∞^𝑛_𝑘=𝑛𝜆_𝑘𝐿𝑢_𝑘≥𝐼̃,𝑝_𝐶𝑐𝛿𝐼̃_𝐶𝑖lim_𝑛→∞𝐿𝑣_𝑛≥𝐼̃,𝑝_𝐶𝑐𝛿𝐼̃_𝐶𝑖lim_𝜀∈∧𝐿𝑣_𝜀,𝑝≥𝐼̃_𝐶𝑐𝛿𝐼̃_𝐶𝑖𝐿𝜔_𝑖,𝑝. (31) Hence, one can obtain that 𝐼̃_𝐶𝑐𝛿𝐼̃_𝐶𝑖𝐺(𝑝₀)≥𝐼̃_𝐶𝑐𝛿𝐼̃_𝐶𝑖𝐿(𝜔_𝑖,𝑝). Because 𝛿 is an arbitrary nonnegative number and 𝐿(⋅,𝑝) has the local property, we have that 𝐼̃_𝐶𝑖𝐺𝑝₀≥𝐼̃_𝐶𝑖𝐿𝜔_𝑖,𝑝, (32) for any 𝑝∈𝐵.
Now, we prove that 𝜔_𝑖=𝐼̃_𝐶𝑖𝑢₀ for any 𝑖∈𝑁. By the definition of 𝑢_𝑛, it is clear that 𝐿𝑢_𝑛,𝑝_𝑛≤𝐿𝑢,𝑝_𝑛,∀𝑢∈𝐴. (33) By condition (2), we have that 1−𝑛⁻¹𝐿𝑢_𝑛,𝑝₀+𝑛⁻¹𝐿𝑢_𝑛,𝑝≤𝐿𝑢,𝑝_𝑛,∀𝑝∈𝐵. (34) Hence, we can obtainthat lim_𝑛→∞1−𝑛⁻¹𝐿𝑢_𝑛,𝑝₀≤lim_𝑛→∞𝐿𝑢,𝑝_𝑛,∀𝑝∈𝐵. (35) Since 𝐿(𝑢_𝑛,𝑝)≥𝐺(𝑝), we can obtain that (1−𝑛⁻¹)𝐿(𝑢_𝑛,𝑝₀)+𝑛⁻¹𝐺(𝑝)≤𝐿(𝑢,𝑝_𝑛) and (1−𝑛⁻¹)𝐿(𝑣_𝑛,𝑝₀)+𝑛⁻¹𝐺(𝑝)≤𝐼̃_𝐶𝑖⋅lim_𝑛→∞((1−𝑛⁻¹)𝐿(𝑢_𝑛,𝑝₀)+𝑛⁻¹𝐿(𝑢_𝑛,𝑝)). According to lim_𝑛→∞𝐿𝑣_𝑛,𝑝₀=lim_𝑛→∞1−𝑛⁻¹𝐿𝑣_𝑛,𝑝₀≤lim_𝑛→∞1−𝑛^−1𝑁𝑛_𝑘=𝑛𝜆_𝑘𝐿𝑢_𝑘,𝑝₀≤lim_𝑛→∞1−𝑛⁻¹𝐿𝑢_𝑛,𝑝₀≤lim_𝑛→∞𝐿𝑢,𝑝_𝑛, (36) where ∑^𝑁𝑛_𝑘=𝑛𝜆_𝑘=1, it is obvious that 𝐼̃_𝐶𝑖𝐿𝜔_𝑖,𝑝₀≤lim_𝑛→∞𝐼̃_𝐶𝑖𝐿𝑣_𝑛,𝑝₀≤lim_𝑛→∞𝐼̃_𝐶𝑖𝐿𝑢_𝑛,𝑝₀≤lim_𝑛→∞𝐼̃_𝐶𝑖𝐿𝑢,𝑝_𝑛. (37) Since {‖𝑝_𝑛−𝑝₀‖,𝑛∈𝑁} converges in probability to 0, by Egoroff theorem, we can obtain that for any 𝜎>0, there exists 𝐶_𝜎∈ℱ such that 𝑃(𝐶_𝜎)<𝜎 and {‖𝑝_𝑛−𝑝₀‖,𝑛∈𝑁} converges to 0 uniformly on 𝐶^𝑐_𝜎. By Remark 37, we can construct a net {𝑝_𝛼∈𝐵,𝛼∈∧} as in Remark 37 such that ‖𝑝_𝛼−𝑝₀‖→0 under the locally 𝐿⁰-convex topology of 𝐸. Hence, by (2) we can obtain that 𝐼̃_𝐶𝑐𝜎𝐼̃_𝐶𝑖𝐿𝜔_𝑖,𝑝₀≤𝐼̃_𝐶𝑐𝜎𝐼̃_𝐶𝑖lim_𝛼∈∧𝐿𝑢,𝑝_𝛼≤𝐼̃_𝐶𝑐𝜎𝐼̃_𝐶𝑖𝐿𝑢,𝑝₀. (38) Since 𝜎 is an arbitrary nonnegative number, it is clear that 𝐼̃_𝐶𝑖𝐿𝜔_𝑖,𝑝₀≤𝐼̃_𝐶𝑖𝐿𝑢,𝑝₀. (39) Therefore, for any 𝑖∈𝑁, 𝜔_𝑖=𝐼̃_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 is 𝑝∈𝐵 such that 𝐿(⋅,𝑝) is not strictly 𝐿⁰(ℱ)-convex on 𝐴, define 𝐿_𝑛𝑢,𝑝=𝐿𝑢,𝑝+𝑛⁻¹‖𝑢‖²,∀𝑛∈𝑁. (40) Since 𝐸 is random strictly convex RN module, we can obtain that 𝐿_𝑛 is strictly 𝐿⁰(ℱ)-convex from Lemma 35. By the similar method, we have that for any 𝑛∈𝑁, there exists (𝑢_𝑛,𝑝_𝑛)∈𝐴×𝐵 such that it is a saddle point of 𝐿_𝑛 with respect to 𝐴×𝐵.
Let 𝜁=⋁{‖𝑦‖∣𝑦∈𝐵} and 𝐷_𝑚 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 {𝐷_𝑚,𝑚∈𝑁} is a countable partition of Ω to ℱ. It is easy to see that {𝐶_𝑖⋂𝐷_𝑗,𝑖,𝑗∈𝑁} is also a countable partition of Ω to ℱ. For any 𝑖,𝑗∈𝑁, let 𝐻_𝑖𝑗=𝐶_𝑖⋂𝐷_𝑗. We can suppose that, without loss of generality, 𝐼̃_𝐻𝑖𝑗𝑢_𝑛∈𝐿²(𝐸) converge to 𝜇_𝑖𝑗 under the weak topology of 𝐿²(𝐸). Then we have that there exists a net {𝐼̃_𝐻𝑖𝑗𝑢_𝛼∈𝐸∣𝛼∈Λ} such that it converges to 𝜇_𝑖𝑗 under the locally 𝐿⁰-convex topology of 𝐸. Thus, we have that 𝐼̃_𝐻𝑖𝑗𝐿𝜇_𝑖𝑗,𝑝≤lim_𝛼𝐼̃_𝐻𝑖𝑗𝐿𝐼̃_𝐻𝑖𝑗𝑢_𝛼,𝑝≤lim_𝑛→∞𝐼̃_𝐻𝑖𝑗𝐿𝐼̃_𝐻𝑖𝑗𝑢_𝑛,𝑝+𝑛⁻¹𝑢_𝑛²≤lim_𝑛→∞𝐼̃_𝐻𝑖𝑗𝐿𝑢,𝐼̃_𝐻𝑖𝑗𝑝_𝑛+𝑛⁻¹𝑢² (41) for all 𝑢∈𝐴,𝑝∈𝐵. Similarly, for any 𝑖,𝑗∈𝑁, one can have a net {𝐼̃_𝐻𝑖𝑗𝑝_𝛽∈𝐸∣𝛽∈Γ} such that it converges to 𝜈_𝑖𝑗 under the locally 𝐿⁰-convex topology of 𝐸 and lim_𝑛→∞𝐼̃_𝐻𝑖𝑗𝐿𝑢,𝐼̃_𝐻𝑖𝑗𝑝_𝑛+𝑛⁻¹‖𝑢‖²≤lim_𝑛→∞𝐼̃_𝐻𝑖𝑗𝐿𝑢,𝐼̃_𝐻𝑖𝑗𝑝_𝑛+lim_𝑛→∞𝑛⁻¹‖𝑢‖²≤lim_𝛽∈Γ𝐼̃_𝐻𝑖𝑗𝐿𝑢,𝐼̃_𝐻𝑖𝑗𝑝_𝛽+lim_𝑛→∞𝑛⁻¹‖𝑢‖²≤𝐼̃_𝐻𝑖𝑗𝐿𝑢,𝜈_𝑖𝑗. (42) Let 𝑢₀=∑_{𝑖,𝑗∈𝑁}𝐼̃_𝐻𝑖𝑗𝜇_𝑖𝑗 and 𝑝₀=∑_{𝑖,𝑗∈𝑁}𝐼̃_𝐻𝑖𝑗𝜈_𝑖𝑗; it is easy to check that 𝐿𝑢₀,𝑝≤𝐿𝑢₀,𝑝₀≤𝐿𝑢,𝑝₀ (43) 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.

Acknowledgment

This work is supported by NNSF no. 11171015.