A basic strict separation theorem in random locally convex modules

Nonlinear Analysis 71 (2009) 3794–3804

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

A basic strict separation theorem in random locally convex modulesI Guo Tiexin a,∗ , Xiao Haixia b , Chen Xinxiang c a

LMIB and Department of Mathematics, Beihang University, Beijing 100191, China

b

Department of Basic Sciences, Hubei Automotive Industries Institute, Shiyan 442002, China

c

School of Mathematical Sciences, Xiamen University, Fujian 361005, China

article

info

Article history: Received 5 August 2008 Accepted 6 February 2009 MSC: 46A22 46A16 46H25 46H05

abstract The central idea of this paper is to make full use of the recently developed theory of random conjugate spaces to establish a basic strict separation theorem that is universally suitable in an arbitrary random locally convex module. A series of interesting corollaries of the basic theorem are also included. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Random locally convex module Continuous canonical module homomorphism M-convex set Strict separation theorem

1. Introduction It is well known that classical separation theorems between two disjoint convex subsets in locally convex spaces have played crucial roles in many important topics in functional analysis [1,2]. Recently, the development of random metric theory in the direction of functional analysis has led to many deep advances in the theory of random normed modules together with their random conjugate spaces [3–10]. These advances, in particular the results of the papers [9,10] lead naturally to the study of random locally convex modules (also called random seminormed modules in [11,12]). Random locally convex modules are a random generalization of locally convex spaces. But except for the classical case when they degenerate to locally convex spaces, they admit hardly any nontrivial continuous linear functionals (see Section 2 of this paper for details). Consequently, classical separation theorems depending on the theory of classical conjugate spaces rarely apply to random locally convex modules (see Remark 2.6). The purpose of this paper is to make full use of the recently developed theory of random conjugate spaces to give several strict separation theorems universally applicable to all random locally convex modules. Precisely speaking, let K be the scalar field of real numbers or complex numbers, (Ω , A, µ) a given σ -finite measure space and L(µ, K ) the algebra of µ-equivalence classes of K -valued µ-measurable functions on Ω . Then L(µ, K ) becomes a Hausdorff topological algebra when endowed with the topology of convergence locally in measure µ. At the same time a random locally convex module over K with base (Ω , A, µ) is just a topological module over the topological algebra L(µ, K ).

I Supported by the National Natural Science Foundation of China (No. 10871016).

∗

Corresponding author. Fax: +86 010 82317934. E-mail addresses: [email protected], [email protected] (T.X. Guo).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.038

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

3795

A central result of this paper is that if A is a closed M-convex (namely, convex in the sense of module, which is stronger than convexity in the sense of a linear space) subset of S and p ∈ S \ A, then there exists a continuous module homomorphism from S to L(µ, K ) strictly separating p from A. A counterexample shows that the M-convexity cannot be replaced by the usual convexity. The above central result can be regarded as a natural generalization of the corresponding classical result. The proof of the classical result can be easily derived from the algebraic (or analytic) form of the classical Hahn-Banach theorem, since the famous Minkowski functionals provide an extremely convenient one-to-one correspondence between open convex neighborhoods and continuous seminorms. Since a random version of the Minkowski functional is not available for random metric theory, we cannot obtain our central result from the algebraic form of the Hahn–Banach theorem previously established for random linear functionals. In fact, the method used in this paper, namely, skillful utilization of the connection between the theory of random conjugate spaces and the theory of classical conjugate spaces, is completely new and more involved than the corresponding classical method. Essentially, our idea is motivated, to some extent, by the paper [9]. The remainder of this paper is organized as follows: In Section 2, we briefly recall some necessary basic notions and facts; then in Section 3 we present and prove our main results.

2. Preliminaries Throughout this paper, (Ω , A, µ) always denotes a given σ -finite positive measure space, we also let µ ˜ denote the P 1 following probability measure: when µ is finite µ( ˜ A) = µ(A)/µ(Ω ), ∀A ∈ A; when µ is not finite µ( ˜ A) = n≥1 2n · µ(A∩An ) , ∀A ∈ A, where {An : n ∈ N } is a chosen countable partition of Ω into A such that 0 < µ(An ) < +∞ for each n ≥ 1, µ(A ) n

and N stands for the set of positive integers. Clearly µ and µ ˜ are equivalent. Further, for terminologies such as µ-measurable functions, µ-almost everywhere (briefly, µ-a.e.) and µ-equivalence classes, we refer to [1]; in particular, L(µ, K ) always denotes the algebra over K of µ-equivalence classes of K -valued µmeasurable functions on Ω , where K stands for the scalar field R of real numbers or C of complex numbers.

Proposition 2.1 ([1]). Denote by L˜ (µ, R) the set of µ-equivalence classes of extended real-valued µ-measurable functions on Ω . L˜ (µ, R) is partially ordered by ξ ≤ η iff ξ 0 (ω) ≤ η0 (ω), µ-a.e. for any chosen representatives ξ 0 and η0 of ξ and η, respectively. Then (L˜ (µ, R), ≤) has the following pleasant properties: W V (1) every subset A of L˜ (µ, R) has a supremum A and an infimum A; W W V V (2) there exist countable subsets {an : n ∈ N } and {bn : n ∈ N } in A such that n≥1 an = A and n≥1 bn = A; (3) If, further, A is directed (dually directed), then the above {an : n ∈ N } (accordingly, {bn : n ∈ N }) can be chosen as nondecreasing (correspondingly, nonincreasing). Furthermore, L(µ, R), as a sublattice of L˜ (µ, R), is also complete in the sense that every subset having an upper bound possesses a supremum. Denote by 0 and 1 the null and unit elements of L(µ, R), respectively, by L˜ + (µ) the set {ξ ∈ L˜ (µ, R) : ξ ≥ 0} and by L+ (µ) the set {ξ ∈ L(µ, R) : ξ ≥ 0}. Remark 2.1. Let B be a subfamily of A and H = {˜IB : B ∈ B }, where ˜IB denotes the µ-equivalence class of the characteristic function IB of B (namely, IB (ω) = 1 if ω ∈ B, and 0 otherwise). Throughout this paper, the notation ˜IB is always understood as above unless stated otherwise. Then by Proposition 2.1(2) there exist countable subfamilies {Bn : n ∈ N } and {Dn : n ∈ N } W W V V S T of B such that {˜IBn : n ∈ N } = {˜IB : B ∈ B }, and {˜IDn : n ∈ N } = {˜IB : B ∈ B }, whence n≥1 Bn and n≥1 Dn are called the essential supremum of B and the essential infimum of B , respectively, and denoted by esssup(B ) and essinf(B ), respectively. Obviously, they are both unique µ-a.e.. Definition 2.1 ([8, Section 7]). An ordered pair (S , {Xd }d∈D ) is called a random locally convex space over K with base (Ω , A, µ) if S is a linear space over K , D is an indexing set and for each d ∈ D, Xd is a mapping from S to L+ (µ) such that the following three conditions are satisfied: W (1) {Xpd : d ∈ D} = 0 iff p = θ (the null vector in S), where Xpd = Xd (p), ∀d ∈ D and p ∈ S; (2) Xαdp = |α| · Xpd , ∀α ∈ K , d ∈ D and p ∈ S;

(3) Xpd+q ≤ Xpd + Xqd , ∀p, q ∈ S and d ∈ D. Furthermore, if there exists another mapping ∗ : L(µ, K ) × S → S such that the following two conditions are also satisfied: (4) (S , ∗) is a left module over the algebra L(µ, K ); (5) Xξd∗p = |ξ | · Xpd , ∀ξ ∈ L(µ, K ), p ∈ S and d ∈ D. Then the triple (S , {Xd }d∈D , ∗) is called a random locally convex module over K with base (Ω , A, µ).

Remark 2.2. (1) The above (4) of Definition 2.1 shows that αˆ ∗ p = (α · 1) ∗ p = α(1 ∗ p) = α · p, ∀α ∈ K and p ∈ S, where αˆ stands for the µ-equivalence class of the constant function with value α , so the module multiplication ∗ can be regarded as a natural extension of the scalar multiplication, and from now on we briefly write (S , {Xd }d∈D ) for (S , {Xd }d∈D , ∗), and ξ · p for

3796

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

ξ ∗ p whenever ∗ is given. (2) In (5) of Definition 2.1, |ξ | stands for the µ-equivalence class of the µ-measurable function |ξ 0 | defined by |ξ 0 |(ω) = |ξ 0 (ω)|, ∀ω ∈ Ω , where ξ 0 is any chosen representative of ξ , so that (5) is a strengthened form of (2) of Definition 2.1. (3) According to the convention from probability theory, the terminology ‘‘random’’ means that (Ω , A, µ) is a probability space. However our work in random metric theory shows that letting (Ω , A, µ) be a σ -finite measure space— which introduces no difficulties—makes it possible for us to apply our results in a straightforward and convenient fashion to problems in functional analysis (see, for example, [13,14]). (4) The terminology ‘‘A random locally convex space’’ was first used by Gong in [15] where Xpd was assumed to be a nonnegative random variable, which is consistent with the idea of [16, Chapter 9]. Here Xpd is assumed to be a µ-equivalence class, namely Xpd is in L+ (µ), consistent with the idea of [5] (see [5] for the reasons and details). (5) In fact, the notion of a random locally convex space in the sense of this paper was employed in [11,12] under the name of ‘‘A random seminormed space’’, the advantage of the latter is to emphasize that the structure of the space is essentially determined by a family of random seminorms, whereas the advantage of the former is to clearly tell the readers that all the results for random locally convex spaces are natural generalizations of those for locally convex spaces, although random locally convex spaces are rarely locally convex spaces. This paper again employs the terminology ‘‘random locally convex spaces’’ in order to express similar information, and we also suggest that this terminology be adopted, since the terminology ‘‘random seminormed space’’ usually stands for ‘‘a space endowed with a single random seminorm’’.

Example 2.1. Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ). If (Ω , A, µ) is a trivial probability space, namely A = {Ω , ∅} and µ(Ω ) = 1, then (S , {Xd }d∈D ) is exactly a locally convex space in the usual sense. Example 2.2. Let (S , {Xd }d∈D ) be a random locally convex module (or, more generally, a random locally convex space) over K with base (Ω , A, µ). When D is a singleton, namely when {Xd }d∈D degenerates to a single mapping X from S to L+ (µ), then (S , X) degenerates to a random normed module (correspondingly, a random normed space), as first introduced in [5]. In particular, L(µ, K ) and L(µ, B) are typical random normed modules, see Example 2.1 of [9] for details. Definition 2.2. Let (S , {Xd }d∈D ) be a random locally convex space over K with base (Ω , A, µ). Denote by µ ˜ the same probability measure as at the beginning of this section. For a given d in D and given positive numbers ε and λ such that λ < 1, let Nθ (d, ε, λ) = {p ∈ S : µ{ω ˜ ∈ Ω : Xpd (ω) < ε} > 1 − λ}, where θ is the null element of S. Then it is easy to see that the family Uθ = { i=1 Nθ (di , εi , λi ) : n ∈ N , and for each 1 ≤ i ≤ n, di ∈ D, εi > 0, 0 < λi < 1} forms a local base at θ of some Hausdorff linear topology for S, called the (ε, λ)-topology for S generated by {Xd }d∈D , or briefly, {Xd }d∈D -topology. It is obvious that a net {pα : α ∈ Γ } converges in the {Xd }d∈D -topology to some p in S iff the net {Xpdα −p : α ∈ Γ }

Tn

converges in measure µ ˜ to 0 for each d ∈ D, equivalently iff the net {Xpdα −p : α ∈ Γ } converges locally in measure µ to 0 for each d in D, namely it converges in measure µ to 0 on each A-measurable A with finite positive measure for each d in D. Thus L(µ, K ), as a random locally convex space, is endowed with the topology of convergence locally in measure, and is easily proved to be a topological algebra over K , namely the algebra multiplication · : L(µ, K ) × L(µ, K ) → L(µ, K ) is jointly continuous. Further if (S , {Xd }d∈D ) is also a random locally convex module, then it is also a topological module over the topological algebra L(µ, K ), namely the module multiplication : L(µ, K ) × S → S is jointly continuous when L(µ, K ) and S are both endowed with their respective (ε, λ)-topologies.

Remark 2.3. When we speak of the topological structure of a random locally convex space, we always mean that it is endowed with its (ε, λ)-topology unless stated otherwise, and the (ε, λ)-topology in this paper is the same as the one introduced in [11, Theorem 2.3]. This kind of (ε, λ)-topology is important because the usual topology of convergence in measure µ is not necessarily a linear topology for L(µ, K ) when µ is not finite: for example, let Ω = R, A = the σ − algebra of Lebesgue measurable subsets of R, µ = the Lebesgue measure on A, and p the µ-equivalence class of the identity function on R. It is easy to see that µ{ω ∈ Ω | 1n |p(ω)| ≥ } = +∞ for each natural number n and each positive number  , so { 1n p : n ∈ N } does not converge to 0 in measure µ, which shows that the topology for L(µ, R) of convergence in measure µ is not a linear topology although it is always a metrizable topology!

The notion of a random conjugate space for a random normed space, which was first introduced in [5], and the work of [12] on the characterization of continuous module homomorphisms on random locally convex modules, together motivate the following: Definition 2.3. Let (S , {Xd }d∈D ) be a random locally convex space over K with base (Ω , A, µ). Then a linear operator f from S to L(µ, K ) (such a linear operator is called a random linear functional on S) is called a µ-a.e. bounded random linear functional on S if there exist countable partition {An : n ∈ N } of Ω into A, a sequence {ξn : n ∈ N } in L+ (µ) and a countable P Fn ˜ subfamily {Fn : n ∈ N } of F (D) such that |f (p)| ≤ n≥1 IAn · ξn · Xp , ∀p ∈ S, where F (D) stands for the family of nonempty finite subsets of D and XF : S → L+ (µ) is defined by XpF =

P

d∈F

Xpd , ∀p ∈ S and F ∈ F (D).

Denote by S ∗ the linear space of all µ-a.e. bounded random linear functionals on S under ordinary addition and scalar multiplication on linear operators, again define ⊗ : L(µ, K ) × S ∗ → S ∗ by (ξ ⊗ f )(p) = ξ · (f (p)), ∀p ∈ S , f ∈ S ∗ and ξ ∈ L(µ, K ), then one can easily see that (S ∗ , ⊗) is a left module over the algebra L(µ, K ), called the random conjugate space for S.

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

3797

Remark 2.4. It is easy to see that a µ-a.e. bounded random linear functional must be continuous, but not conversely. In fact, the notion of a continuous random linear functional is too weak to guarantee that any meaningful results can be obtained. The following Lemma 2.1 exhibits the substance of a µ-a.e. bounded random linear functional. Lemma 2.1 ([12]). Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ). Then f ∈ S ∗ iff f is a continuous module homomorphism from S to L(µ, K ). The following two propositions, which are due to Guo [5] together with their rigorous proofs [7], show that any random locally convex space admits enough µ-a.e. bounded random linear functionals, and any random locally convex module enough continuous canonical module homomorphisms. Proposition 2.2 ([5,7]). Let S be a real linear space, M a subspace of S, f : M → L(µ, R) a random linear functional and X : S → L(µ, R) a random sublinear functional (namely, it is positively homogeneous and subadditive) such that f (p) ≤ Xp , ∀p ∈ M. Then there exists a random linear functional F : S → L(µ, R) such that F extends f and F (p) ≤ Xp , ∀p ∈ S. Proposition 2.3 ([5,7]). Let S be a linear space over K , M a subspace of S, f : M → L(µ, K ) a random linear functional and X : S → L+ (µ) a random seminorm such that |f (p)| ≤ Xp , ∀p ∈ M. Then there exists a random linear functional F : S → L(µ, K ) such that F extends f and |F (p)| ≤ Xp , ∀p ∈ S. The following Definition 2.4 and Proposition 2.4 show that the theory of random conjugate spaces is crucial for random locally convex modules, which, when further combined with Proposition 2.3, lead us to the idea of this paper. Definition 2.4. Let (S , {Xd }d∈D ) be a random locally convex space over K with base (Ω , A, µ). Let ξ =

W

{Xpd : p ∈ S and d ∈

D}, and ξ any chosen representative of ξ . Then µ{ω ∈ Ω : 0 < ξ (ω) < +∞} = 0 since S is a linear space. The set {ω ∈ Ω : ξ 0 (ω) > 0} is called a support of S. Clearly the supports of S are µ-a.e. unique. If Ω is a support of S, then S is said to have full support. 0

0

Remark 2.5. Any random locally convex space (S , {Xd }d∈D ) over K with base (Ω , A, µ) can always be made into one having full support: Let Ω0 ∈ A be a support of S (such a support always exists!), A0 = Ω0 ∩ A = {Ω0 ∩ A : A ∈ A} and µ0 = ˆ d : S → L+ (µ0 ) by Xˆ pd = the restriction of Xpd to Ω0 . Then (S , {X ˆ d }d∈D ) is the restriction of µ to A0 . For each d in D, define X a random locally convex space over K with base (Ω0 , A0 , µ0 ), which has full support and is essentially identified with the original (S , {Xd }d∈D ). Proposition 2.4 ([17]). Let (S , {Xd }d∈D ) be a random locally convex module of full support over K with base (Ω , A, µ). Then we can have the following statements: (1) S admits a nontrivial continuous linear functional (equivalently, a proper open convex subset) iff A has a µ-atom (see [1] for the definition of µ-atoms); (2) S admits enough nontrivial continuous linear functionals iff A is essentially generated by countably many disjoint µ-atoms; (3) S is locally convex iff A is essentially generated by countably many disjoint µ-atoms. Remark 2.6. Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ). As pointed out in [10], pA = ˜IA · p is called the A-stratification of p for each given µ-measurable set A of Ω and p in S, and clearly, S is stratified, namely, S includes every stratification of an element in S. Clearly, pA = θ when µ(A) = 0, and pA = p when µ(Ω \ A) = 0, and both are called trivial stratifications of p. Further, when (Ω , A, µ) is a trivial probability space every element in S has merely the two trivial stratifications since A = {Ω , ∅}; when (Ω , A, µ) is arbitrary, every element in S can possess arbitrarily many nontrivial intermediate stratifications. Proposition 2.4 shows that this kind of complicated stratification structure of an arbitrary random locally convex module leads to the universal failure of classical conjugate space theory. In fact, it is also the essence of our work in this paper that many difficulties coming from the study of stratification structure can be overcome. This will be reflected, in particular, in Lemma 3.2 and Theorem 3.1 of this paper together with their proofs. Obviously the case in which A is essentially generated by countably many disjoint µ-atoms is a quite extremal one, since there is no randomness to speak of. Thus within random metric theory, almost all classical separation theorems of two disjoint nonempty convex subsets fail to meet the needs of the theory of random locally convex modules since these theorems essentially require that one of the two convex sets is open. Definition 2.5. Let S be a left module over the algebra L(µ, K ) and A a subset of S. Then A is called M-convex (namely, convex in the sense of a module) if ξ · p + η · q still belongs to A whenever p, q are any given elements in A and ξ and η in L+ (µ) are such that ξ + η = 1; A is called M-balanced if ξ · p ∈ A for a given p in A, and ξ ∈ L(µ, K ) such that |ξ | ≤ 1. An M-convex set must be convex, but not conversely. The pleasant properties of an M-convex set are reflected in the following Lemma 2.2, which is essential for the proof of our main results. For the sake of convenience and also for Section 3 of this paper, we introduce Definition 2.6 below before stating Lemma 2.2.

3798

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

Definition 2.6. (1) For ξ and η in L(µ, R), let ξ 0 and η0 be arbitrarily chosen representatives of ξ and η, respectively, and A = {ω ∈ Ω : ξ 0 (ω) ≤ η0 (ω)}, then we usually write [ξ ≤ η] for the µ-equivalence class of A, and I[ξ ≤η] for ˜IA , similarly one can easily understand notations such as I[ξ >η] and I[ξ =η] . (2) For ξ ∈ L(µ, K ) with a representative ξ 0 , the function Q (ξ 0 ) defined by Q (ξ 0 )(ω) = 1/ξ 0 (ω) if ξ 0 (ω) 6= 0, and 0 otherwise, is called the generalized inverse of ξ 0 , and the µ-equivalence class Q (ξ ) of Q (ξ 0 ) is called the generalized inverse of ξ , clearly Q (ξ ) · ξ = ξ · Q (ξ ) = I[ξ 6=0] . (3) For any given ξ in L(µ, C ), if ξ = ξ1 + iξ2 , ξ1 , ξ2 ∈ L(µ, R), then ξ¯ = ξ1 − iξ2 is called the complex conjugate of ξ . From now on, for a given random locally convex module (S , {Xd }d∈D ) over K with base (Ω , A, µ) , F (D) always denotes P d the family of nonempty finite subsets of D. For each F ∈ F (D), XF is always defined by XpF = d∈F Xp , ∀p ∈ S. Then

(S , {F F }F ∈F (D) ) is also a random locally convex module over K with base (Ω , A, µ). First, each XF is continuous with respect to the {Xd }d∈D -topology for S, and {XF : F ∈ F (D)} includes {Xd : d ∈ D}. Thus {XF : F ∈ F (D)} and {Xd : d ∈ D} generate the same (ε, λ)-topology. Secondly, F (D) is directed by the set inclusion relation : E ≤ F iff E ⊂ F , and it is also obvious that if E ⊂ F , then XpE ≤ XpF , ∀p ∈ S, so (S , {XF }F ∈F (D) ) is exactly a random seminormed module in the sense of [11, Definition 2.4]. The above observations lead us to:

Lemma 2.2. Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ), p ∈ S, and A an M-convex subset of S. Then we have: (1) {XpF−q : q ∈ A} is both directed and dually directed for each F in F (D);

(2) If A is also closed, then p ∈ A iff

W

{ξF : F ∈ F (D)} = 0, where ξF =

V

{XpF−q : q ∈ A}, for each F ∈ F (D).

Proof. When A is a submodule of S, by applying [11, Lemma 3.2] to (S , {XF }F ∈F (D) ) one can easily see that (1) holds; by further noticing that the proof of [11, Lemma 3.2] used only the M-convexity of the submodule, one can also see that (1) always holds. Since A is a closed M-convex set in (S , {Xd }d∈D ) iff A is a closed M-convex set in (S , {XF }F ∈F (D) ), by applying [11, Lemma 3.3] to (S , {XF }F ∈F (D) ) and by noticing that the method of the proof of [11, Lemma 3.3] remains valid when the closed submodule there is replaced by a closed M-convex set, one can easily see that (2) always holds.  Finally, we conclude this section with Lemma 2.3 and Proposition 2.5 below, which are key to the proof of our central result. Before giving them, we introduce some convenient notations. Although we have given the definition of a random normed module (briefly, an RN module) in an indirect way through Example 2.2, we suggest the readers to refer [5] or [9, Definitions 2.1 and 2.2] for the concrete definitions of RN modules and their random conjugate spaces. For a given RN module (S , X) over K with base (Ω , A, µ), and a given positive or extended nonnegative real number 1

p such that 1 ≤ p ≤ +∞, define k · kp : S → [0, +∞] by kg kp = ( Ω (Xg )p dµ) p if 1 ≤ p < +∞, and the µ-essential supremum of Xg if p = +∞. Let Lp (S ) = {g ∈ S : kg kp < +∞}. Since the random conjugate space (S ∗ , X∗ ) of (S , X) is also a complete RN module, we also have the corresponding Lq (S ∗ ) for any q such that 1 ≤ q ≤ +∞.

R

Lemma 2.3. Let (S , X) be an RN module over K with base (Ω , A, µ), and G an M-convex subset of S including θ . Then we have the following statements: (1) For each p such that 1 ≤ p ≤ +∞, (Lp (S ), k · kp ) is a normed space over K , and is further a Banach space if (S , X) is complete; (2) For each p such that 1 ≤ p ≤ +∞, Lp (G) is also an M-convex set in S, and is dense in G with respect to the (ε, λ)-topology, in particular Lp (S ) is dense in S, where Lp (G) = {g ∈ G : kg kp < +∞}; (3) Lp (G) is closed in the norm topology in Lp (S ) if G is closed in S in the (ε, λ)-topology. Proof. We only need to prove Lp (G) is dense in G in the (ε, λ)-topology, and the rest follows from a straightforward verification. Let g be a given element in G, An = [Xg ≤ n] and gn = IAn · g for each n ∈ N. We can, without loss of generality, assume (Ω , A, µ) is a finite measure space (when it is σ -finite, the proof only needs a slight but tedious modification). Then gn ∈ G since gn = IAn · g + (1 − IAn ) · θ and θ ∈ G, we have gn ∈ Lp (G). Since Xg −gn = (1 − IAn )Xg converges µ-a.e. to 0 as n → ∞, then {Xg −gn : n ∈ N } also converges in measure µ to 0, namely {gn : n ∈ N } converges in the (ε, λ)-topology to g.  Proposition 2.5 ([4] see also [8, Theorem 6.1] for a proof in English). Let (S , X) be an RN module over K with base (Ω , A, µ), and (S ∗ , X∗ ) the random conjugate space for (S , X) (notice that (S ∗ , X∗ ) is a complete RN module over K with base (Ω , A, µ)). For a given positive number p such that 1 ≤ p < +∞, q denotes the Hölder conjugate number of p. Then (Lp (S ))0 is isomorphically isometric onto Lq (S ∗ )Runder the canonical mapping T : Lq (S ∗ ) → (Lp (S ))0 defined as follows: for each f ∈ Lq (S ∗ ), Tf : Lp (S ) → K is given by Tf (g ) = Ω f (g ) dµ, ∀g ∈ Lp (S ), where Tf denotes T (f ), and (Lp (S ))0 is the classical conjugate space of Lp (S ).

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

3799

3. Main results and proofs The central result of this paper is Theorem 3.1 below, whose proof is long, and thus divided into three lemmas. To state the proof of Theorem 3.1 together with several corollaries conveniently and concisely, we again introduce some notation in Definition 3.1 below and also remind the readers of the notations in Definition 2.6. Definition 3.1. Let Aˆ µ be the Lebesgue completion of A with respect to µ (namely the σ -algebra of µ-measurable sets of Ω , see [1] for details), and A(µ) the set of µ-equivalence classes of µ-measurable sets, namely A(µ) = {A˜ : A ∈ Aˆ µ }, where A˜ denotes the µ-equivalence class of A (that is, A˜ = {B ∈ Aˆ µ : µ(B4A) = 0}, where B4A denotes the symmetric difference of B and A). Then we make the following conventions: (1) A˜ 4B˜ stands for the µ-equivalence class of A4B, for any A˜ and B˜ in A(µ), in fact, each set operation on Aˆ µ induces a corresponding reasonable one on A(µ), and one can easily understand the other operations as above; (2) µ(A˜ ) is defined to be µ(A) for each A ∈ Aˆ µ ; (3) IA˜ denotes ˜IA , where ˜IA stands for the µ-equivalence class of IA , and IA is the characteristic function of A, for each A ∈ Aˆ µ ; (4) A˜ ⊂ B˜ iff IA˜ ≤ IB˜ , the latter being understood as in Proposition 2.1, this means A˜ ⊂ B˜ iff IA (ω) ≤ IB (ω), µ-a.e., also iff A ⊂ B, µ-a.e.. Remark 3.1. If (Ω , A, µ) is finite, define d : A(µ) × A(µ) → A(µ) by d(A˜ , B˜ ) = µ(A˜ 4B˜ ) = µ(A4B), ∀A, B ∈ Aˆ µ . Then (A(µ), d) is a complete metric space, which has frequently been employed in measure theory. If {An : n ∈ N } is a sequence S P P ˜ of disjoint sets in A, then one can easily see that µ( n≥1 A˜ n ) = n≥1 µ(An ) = n≥1 µ(An ). Theorem 3.1. Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ), (S , {XF }F ∈F (D) ) the one associated with (S , {Xd }d∈D ) as in the paragraphWbefore Lemma 2.2, G a closed M-convex subset of S, p0 ∈ S \ G, ξF = V F {Xp0 −g : g ∈ G} for each F ∈ F (D), and ξ = {ξF : F ∈ F (D)}. Then there exists a continuous module homomorphism f from S to L(µ, K ) such W that f has the following two properties: (1) (Ref )(p0 ) > {(Ref )(g ) : g ∈ G}, where Ref denotes the real part of f , namely (Ref )(p) is the real part of f (p) and it is easy to verify that f (p) = (Ref )(p) − i(Ref )(ip), ∀p ∈ S, in addition, ‘‘ >’’W means ‘‘ ≥’’ but ‘‘ 6=’’; (2) µ(E 4E 0 ) = 0, where E = [ξ > 0] and E 0 = [(Ref )(p0 ) > {(Ref )(g ) : g ∈ G}] are understood according to Definition 2.6. Remark 3.2. If (Ω , A, µ) is a probability space, then µ(E ) is interpreted as the probability that p0 does not belong to G, namely that p0 is, originally, topologically away from G; µ(E 0 ) as the probability that f separates p0 from G. The above (2) of Theorem 3.1 shows that f can be chosen so that the separation probability of p0 from G by f can be as high as the probability that p0 is topologically away from G. Generally whenever f satisfies (1) of Theorem 3.1, then since (1 − IE ) · ξ = 0 (1 denotes the unit element of L(µ, K )), namely IE c · ξ = 0 (if E = E˜ 0 , then E c = Ee0c , E0c = Ω \ E0 , see Definition 3.1 for details), this means that IE c · p0 ∈ IE c · G = {IE c · g : g ∈ G}, so E 0 ⊂ E must hold; and (2) of Theorem 3.1 just requires E 0 = E! Such probabilistic interpretations as above come from the idea of [16, Section 10 of Chapter 12]. Before giving the proof of Theorem 3.1 together with the three lemmas that lead up to it, we first state and prove three corollaries of Theorem 3.1. Corollary 3.1 ([1]). Let S be a locally convex space over K whose topology is generated by a family {Pd : d ∈ D} of seminorms on S, G a closed convex set of S and p0 ∈ S \ G. Then there exists a continuous linear functional f on S such that (Ref )(p0 ) > sup{(Ref )(g ) : g ∈ G}. Proof. In Theorem 3.1, take (Ω , A, µ) to be the trivial probability space, namely A = {Ω , ∅} and µ(Ω ) = 1. Then (S , {Xd }d∈D ) just degenerates to a locally convex space in the usual sense, e.g., (S , {Pd }d∈D ), a closed M-convex set to a closed convex set in the usual sense, a continuous module homomorphism to a continuous linear functional in the usual sense. Then (1) of Theorem 3.1 is our desired conclusion. By the way, (2) in Theorem 3.1 also holds automatically upon noticing that in this case E and E 0 are both Ω .  Corollary 3.2. Let (S , {Xd }d∈D ) and (S , {XF }F ∈F (D) ) be as in Theorem 3.1, A and B be two disjoint closed M-convex subsets of S such that A is compact. Then there exists a continuous module homomorphism f from S to L(µ, K ) such that f satisfies the following V two conditions: W (1) {(Ref )(b) : b ∈ B} > {(Ref )(a) : a ∈ A}; W V F (2) µ(U 4V ) = 0, V where U = [ξ > 0] and V = [η W> γ ], and ξ = {ξF : F ∈ F (D)}, ξF = {Xb−a : b ∈ B and a ∈ A} for each F ∈ F (D), η = {(Ref )(b) : b ∈ B} and γ = {(Ref )(a) : a ∈ A}.

3800

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

Proof. Let G = B − A = {b − a : b ∈ B and a ∈ A}. Then G is a closed M-convex set since A and B are closed M-convex subsets and A is compact, and θ 6∈ G since A ∩ B = ∅. Take p0 = θ and G = B − A in Theorem 3.1. Then ξ and {ξF : F ∈ F (D)} are exactly the same as in Theorem 3.1. Denote f˜ as obtained inV Theorem 3.1 with respect to θ and G, then f = −f˜ will satisfy our requirements. First, h =: {(Ref )(b −W a) : b ∈ B and a ∈ A} > 0, and hence (Ref )(b) > (Ref )(a), ∀b ∈ B and a ∈ A, whence V η = {(Ref )(b) : b ∈ B} ≥ {(Ref )(a) : a ∈ A} = γ . Second, {(Ref )(a) : a ∈ A} is directed: in fact, let H = [(Ref )(a1 ) ≤ (Ref )(a2 )] for any given a1 , a2 in A. Then (Ref )(a1 ) ∨ (Ref )(a2 ) = IH · (Ref )(a2 ) + (1 − IH )(Ref )(a1 ) = (Ref )(a3 ), since f is a module homomorphism, where a3 = (1 − IH ) · (a1 ) + IH · a2 , one can easily see that a3 ∈ A (notice A is M-convex), this shows that {(Ref )(a) : a ∈ A} is, of course, directed. Thus by Proposition 2.1 there exists a sequence {an : n ∈ N } in A such that {(Ref )(an ) : n ∈ N } % γ . Further, since A is compact, there exists a subnet {˜aν : ν ∈ Γ } of {an : n ∈ N }, which is convergent to some a0 ∈ A, and hence γ = (Ref )(a0 ) by the continuity of f . V To complete the proof of (2), we only need to verify that h = {(Ref )(b − a) : b ∈ B and a ∈ A} = η − γ . Since [h > 0] = [η > γ ] =: V and since Theorem 3.1 shows that µ(U 4[h > 0]) = 0, we will have µ(U 4V ) = 0 once we can verify h = η − γ . This will proceed as follows. Completely similar to the proof that {(Ref )(a) : a ∈ A} is directed, one can see that {(Ref )(b − a) : b ∈ B and a ∈ A} and {(Ref )(b) : b ∈ B} are both dually directed. Then by Proposition 2.1 there exist a sequence {bn − an : n ∈ N } in B − A and a sequence {b0n : n ∈ N } in B such that the following three conditions are satisfied: (1)0 : bn ∈ B and an ∈ A for each n ≥ 1; (2)0 : {(Ref )(bn − an ) : n ≥ 1} & h; (3)0 : {(Ref )(b0n ) : n ≥ 1} & η. Then (1)0 and (2)0 imply h ≥ η − γ since (Ref )(bn − an ) = (Ref )(bn ) − (Ref )(an ) ≥ (Ref )(bn ) − γ ≥ η − γ for each n ≥ 1. Let a0 obtained as in the above fourth V paragraph of the proof of this corollary. Then (3)0 implies η − γ = limn→∞ (Ref )(b0n ) − γ = limn→∞ (Ref )(b0n − a0 ) ≥ {(Ref )(b − a) : b ∈ B and a ∈ A} = h. To sum up, we have proved h = η − γ , which also implies η > γ since h > 0, whence (1) is established.  Corollary 3.3. In Theorem 3.1, if G is also M-balanced, then f also has the following two properties: W (1)0 |f (p0 )| > {|f (g )| : g ∈ G}; W (2)0 µ(E 4E 00 ) = 0, where E is the same one as in Theorem 3.1, and E 00 = [|f (p0 )| > {|f (g )| : g ∈ G}]. Proof. Denote {(Ref )(g ) : g ∈ G} by η1 and {|f (g )| : g ∈ G} by η2 , then, clearly, η2 ≥ η1 . On the other hand, we can also have η1 ≥ η2 . In fact, for each given g in G, let sgn(f (g )) = f (g ) · Q (|f (g )|), then sgn(f (g )) · (f (g )) = |f (g )|2 W · Q (|f (g )|) = |f (g )| · I[|f (g )|6=0] = |f (g )|, namely f (sgn(f (g )) · g ) = |f (g )|. This shows that |f (g )| = (Ref )(sgn(f (g )) · g ) ≤ {(Ref )(g 0 ) : g 0 ∈ G} = η1 since |sgn(f (g ))| ≤ 1 and G is M-balanced (and hence sgn(f (g )) · g ∈ G). Since g is arbitrary, we have η2 ≤ η1 , whence η2 = η1 . Finally, |f (p0 )| ≥ (Ref )(p0 ) > η1 (by Theorem 3.1(1))= η2 , so (1)0 holds. Likewise, |f (p0 )| ≥ (Ref )(p0 ) also implies E 00 = [|f (p0 )| > η2 ] ⊃ [(Ref )(p0 ) > η1 ] = E 0 , where E 0 is the same one as in Theorem 3.1. Similar to the assertion that E 0 ⊂ E in Remark 3.2, one can also see that E 00 ⊂ E, so by Theorem 3.1(2), E = E 0 . Thus, finally, E = E 0 ⊂ E 00 ⊂ E, i.e., E = E 0 = E 00 , which shows that (2)0 must hold. 

W

W

We now return to the proof of Theorem 3.1, which will be derived from the following three lemmas. In the sequel of this section we can, without loss of generality, assume that (Ω , A, µ) is always a probability space; otherwise we can consider the associated probability space (Ω , A, µ) ˜ in place of (Ω , A, µ) (see the beginning of Section 2). Lemma 3.1. Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ), {fn : n ∈ N } a sequence in S ∗ , and P {An : n ∈ N } a sequence of disjoint µ-measurable sets of Ω . Then f : S → L(µ, K ) defined by f (p) = n≥1 ˜IAn ·(fn (p)), ∀p ∈ S, is again a continuous module homomorphism, namely, f ∈ S ∗ .

˜ Proof. First, for each p in S, the series P P n≥1 IAn · (fn (p)) always converges µ-a.e. in L(µ, K ), and hence f is well defined. Secondly, since n≥1 µ(An ) = µ( n≥1 An ) ≤ µ(Ω ) = 1, one can easily verify that f is a continuous module homomorphism.  P

For a given RN module (S , X), G is a closed M-convex subset of S, and p0 ∈ S \ G. Since {Xp0 −g : g ∈ G} is dually directed, V one can easily see {Xp0 −g : g ∈ G} > 0 since p0 6∈ G and G is closed; in practice, the assertion can also be regarded as a special case of Lemma 2.2 by taking Xd to be X for each d ∈ D. Lemma 3.2. Let (S , X) be an RN module over K with base (Ω , A, µ), G a closed M-convex subset of S, p0 ∈ S \ G, ξ =: V {Xp0 −g : g ∈ G}, E = [ξ > 0] and E0 an arbitrarily chosen representative of E. Then for each µ-measurable subset H of E0 with µ(H ) > 0, there always exists a µ-measurable subset L of H with µ(L) > 0, and an fH ∈ S ∗ such that the following conditions are satisfied: W (1) (RefH )(p0 ) > {(Ref WH )(g ) : g ∈ G}; (2)e L ⊂ [(RefH )(p0 ) > {(RefH )(g ) : g ∈ G}] ⊂ e H, wheree L and e H denotes the µ-equivalence classes of L and H, respectively.

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

3801

Proof. Without loss of generality, we can think θ ∈ G (otherwise, by a translation). Let An = [Xp0 ≤ n] for each n ≥ 1, then S e (the {An : n ∈ N } ⊂ A(µ) (see Definition 3.1 for A(µ)) with An ⊂ An+1 for each n ≥ 1. Since Xp0 ∈ L+ (µ), n≥1 An = Ω H ) > 0. For simplicity, we can µ-equivalence class of Ω ). Since µ(H ) > 0, there exists at least an n0 ≥ 1 such that µ(An0 ∩ e take n0 =S1. Then µ(An ∩ e H ) > 0 for each n ≥ 1. S Since n≥1 (An ∩ e H) = e H, we can choose a representative Hn from each An ∩ e H such that n≥1 Hn = H and Hn ⊂ Hn+1 . Putting BnS = Hn \ Hn−1 for each n ≥ 1, with H0 = ∅, yields that {Bn : n ∈ N } is a sequence of disjoint µ-measurable sets such that n≥1 Bn = H and Bn ⊂ Hn . Without loss of generality, we can take µ(Bn ) > 0 for each n ≥ 1 since µ(H ) > 0 (otherwise, we can remove it). V V For any given n ≥ 1, take pn0 = ˜IBn · p0 and Gn = ˜IBn · G =: {˜IBn · g : g ∈ G}. Then {Xpn −g : g ∈ Gn } = ˜IBn · ( {Xp0 −g : 0

g ∈ G}) = ˜IBn · ξ > 0 since Bn ⊂ Hn ⊂ H ⊂ E0 and µ(Bn ) > 0. Further, since Gn is also a closed M-convex subset of S, pn0 6∈ Gn for each n ≥ 1. Clearly, pn0 ∈ L2 (S ) \ L2 (Gn ), and L2 (Gn ) is a closed convex subset of L2 (S ) by Lemma 2.3. For each n ≥ 1, applying the classical separation theorem (namely, Corollary 3.1) to pn0 and L2 (Gn ) will produce an Fn ∈ (L2 (S ))0 such that the following relation holds:

(ReFn )(pn0 ) > sup{(ReFn )(g ) : g ∈ L2 (Gn )}.

(3)

Proposition 2.5 will again produce an fn0 ∈ L2 (S ∗ ) such that Fn (g ) = obviously holds:

(ReFn )(g ) =

Z Ω

(Refn0 )(g ) dµ,

R

f 0 (g ) dµ, ∀g Ω n

∀g ∈ L2 (S ).

∈ L2 (S ), then the following also

(4)

(3) and (4) will yield the following relation:

Z Ω

(Refn0 )(pn0 ) dµ > sup

Z Ω

 (Refn0 )(g ) dµ : g ∈ L2 (Gn )

for each n ≥ 1.

(5)

Denote {(Refn0 )(g ) : g ∈ L2 (Gn )} by Tn for each n ≥ 1. Since L2 (Gn ) is an M-convex set by Lemma 2.3, it is also easy to check (as in the proof of Corollary 3.2(1)) that Tn is directed. W Then Proposition 2.1 provides a sequence {gn,k : k ∈ N } in L2 (Gn ) forR each n ≥ 1 such that {(Refn0 )(gn,k ) : k ∈ N } % Tn as k → ∞. Since Ω |(Refn0 )(gn,k )| dµ < +∞, Levy’s convergence theorem guarantees the following:

Z lim

k→∞

(Refn )(gn,k ) dµ = 0

Ω

Z _ Ω

Tn



dµ, for each given n ≥ 1.

(6)

From (5) and (6) one can easily obtain the following relations:

Z Ω

(Refn0 )(pn0 ) dµ > sup

Z

(Refn0 )(g ) dµ : g ∈ L2 (Gn )



ZΩ ≥ lim (Refn0 )(gn,k ) dµ k→∞ Ω Z _  = {(Refn0 )(g ) : g ∈ L2 (Gn )} dµ. Ω

= ˜IBn · p0 and Gn = ˜IBn · G, and by observing g = ˜IBn · g for each g in Gn , we can have the following relations: Z ˜IBn · (Refn0 )(pn0 ) dµ (Refn0 )(pn0 ) dµ = Bn Ω Z = (Refn0 )(˜IBn · pn0 ) dµ Ω Z = (Refn0 )(pn0 ) dµ Ω Z _  > {(Refn0 )(g ) : g ∈ L2 (Gn )} dµ ZΩ  _  = {(Refn0 )(˜IBn · g ) : g ∈ L2 (Gn )} dµ ZΩ _  ˜IBn · = {(Refn0 )(g ) : g ∈ L2 (Gn )} dµ

By recalling pn0

Z

Ω

3802

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

Z _  = {(Refn0 )(g ) : g ∈ L2 (Gn )} dµ.

(7)

Bn

Since L2 (Gn ) is dense in Gn with respect to the (ε, λ)-topology on S, and at the same time fn0 is continuous, one can easily W W verify that {(Refn0 )(g ) : g ∈ Gn } = {(Refn0 )(g ) : g ∈ L2 (Gn )}. Then (7) shows that the following relation also holds:

Z

(Refn0 )(pn0 ) dµ > Bn

Z _

 {(Refn0 )(g ) : g ∈ Gn } dµ for each n ≥ 1.

(8)

Bn

Define fn by fn (p) = ˜IBn · (fn0 (p)), ∀p ∈ S, namely fn = ˜IBn · fn0 for each n ≥ 1. Then fn , clearly, belongs to S ∗ . It is also

easy to see that fn (p0 ) = ˜IBn · (fn0 (p0 )) = fn0 (˜IBn · p0 ) = fn0 (pn0 ), and fn (g ) = ˜IBn · (fn0 (g )) = fn0 (˜IBn · g ), ∀g ∈ G. Thus

W W W {(Refn0 )(g ) : g ∈ Gn } = {(Refn0 )(˜IBn · g ) : g ∈ G} = {(Refn )(g ) : g ∈ G}. This and (8) together imply the following relation:

Z

Z _  (Refn )(p0 ) dµ > {(Refn )(g ) : g ∈ G} dµ for each n ≥ 1. Bn

(9)

Bn

(9) must show that 0, for each n ≥ 1, such that W there always exists a µ-measurable subset Ln of Bn with µ(Ln ) > W e Ln ⊂ [(Refn )(p0 ) > {(Refn )(g ) : g ∈ G}]. According to the definition of fn , [(Refn )(p0 ) > {(Refn )(g ) : g ∈ G}] ⊂ e Bn . Here, e Ln and B˜ n denote the µ-equivalence classes of Ln and Bn , respectively. P ∗ ˜ Finally, define fH : S → L(µ, K ) by fH (p) = n≥1 IBn · (fn (p)), ∀p ∈ S. Then Lemma 3.1 shows fH ∈ S . Now, take S S S W e L = {(Refn )(g ) : g ∈ n≥1 Ln , then L ⊂ n≥1 Bn = n≥1 Hn = H and µ(L) > 0. By noticing that Ln ⊂ [(Refn )(p0 ) > W S W G}] ⊂ e Bn , and [(RefH )(p0 ) > {(RefH )(g ) : g ∈ G}] = n≥1 (e Bn ∩ [(Refn )(p0 ) > {(Refn )(g ) : g ∈ G}]), we have that W e L ⊂ [(RefH )(p0 ) > {(RefH )(g ) : g ∈ G}] ⊂ e H. Thus (2) has been proved, whence our (1) also follows since µ(L) > 0.  Remark 3.3. Theorem 2.1 of [18] is equivalent to the special case of our Lemma 3.2(1) when H = E0 , and our method here also considerably improves that used there by beginning with a decomposition of H to {Bn : n ∈ N }, rather than concluding with a decomposition of H, which makes our proof here more readable. On the other hand, the generality of our Lemma 3.2 will play a crucial role in establishing (2) of Theorem 3.1. Lemma 3.3. Let (S , {Xd }d∈D ), (S , {XF }F ∈F (D) ), p0 , G, {ξF : F ∈ F (D)}, ξ and E be the same ones as in Theorem 3.1. Then for an arbitrarily chosen representative E0 of E and a µ-measurable subset H of E0 with µ(H ) > 0, there always exist a µ-measurable subset L of H with µ(L) > 0 and an fH ∈ S ∗ such that the following conditions are satisfied: W (1) (RefH )(p0 ) > {(RefH )(g ) : g ∈ G};

(2) L˜ ⊂ [(RefH )(p0 ) >

W {(RefH )(g ) : g ∈ G}] ⊂ e H, where e L and e H are interpreted as in Lemma 3.2.

Proof. Let us recall that ξF = {XpF0 −g : g ∈ G} for each F ∈ F (D), and ξ = {ξF : F ∈ F (D)}. First {ξF : F ∈ F (D)} is directed: in fact, for any F1 , F2 ∈ F (D), take F3 = F1 ∪ F2 , then ξF3 ≥ ξF1 ∨ ξF2 . Thus, by Proposition 2.1 there exists a nondecreasing sequence {ξFn : n ∈ N } in {ξF : F ∈ F (D)} such that {ξFn : n ∈ N } % ξ .

V

W

Denote [ξFn > 0] by En for each n ≥ 1. Then

e n≥1 En = [ξ > 0] = E. Since E0 = E , H ⊂ E0 and µ(H ) > 0, there must e be some n0 ≥ 1 such that µ(En0 ∩ H ) > 0. Obviously, we can choose a representative H1 of En0 ∩ e H such that H1 ⊂ H. V Denote XFn0 by X for simplicity. Then ξFn0 = {Xp0 −g : g ∈ G}, and, clearly ξFn0 > 0. Since (S , X) need not become an S

RN module, we will consider its quotient space as follows.

ˆ then ˆ : Sˆ → L+ (µ) by Xˆ pˆ = Xp , ∀ˆp ∈ S, Let Sˆ = {ˆp : p ∈ S }, where pˆ = {q ∈ S : Xp−q = 0} for each p ∈ S. Define X ˆ ) becomes an RN module over K with base (Ω , A, µ). (Sˆ , X V Let J : S → Sˆ be the canonical quotient mapping, namely pˆ = J (p), ∀p ∈ S. Obviously, Xˆ J (p0 ),J (G) =: {Xˆ J (p0 )−J (g ) : g ∈ V ˆ ), which is a closed M-convex G} = {Xp0 −g : g ∈ G} > 0. Thus J (p0 ) 6∈ J (G), where J (G) denotes the closure of J (G) in (Sˆ , X subset. ˆ ∗ ) such that the following two conditions satisfied: Applying Lemma 3.2 to J (p0 ) and J (G) for H1 yields an F ∈ (Sˆ ∗ , X (1) (ReF )(J (p0 )) >

W {(ReF )(ˆg ) : gˆ ∈ J (G)}; W e1 , where L is the associated subset of H1 with µ(L) > 0. (2) e L ⊂ [(ReF )(J (p0 )) > {(ReF )(ˆg ) : gˆ ∈ J (G)}] ⊂ H d ∗ ˆ ). Once Take fH = F ◦ J, then fH ∈ (S , {X }d∈D ) since J is a continuous module homomorphism from (S , {Xd }d∈D ) to (Sˆ , X W W W W we observe that {(ReF )(ˆg ) : gˆ ∈ J (G)} = {(ReF )(ˆg ) : gˆ ∈ J (G)} = {(ReF )(J (g )) : g ∈ G} = {(RefH )(g ) : g ∈ G}, we can see that L and fH are as desired.



T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

3803

Now, we can prove Theorem 3.1 as follows: Proof of Theorem 3.1. Take H = E0 . W Then Lemma 3.3 shows there always exists an f ∈ S ∗ such that (Ref )(p0 ) > W {(Ref )(g ) : g ∈ G} and [(Ref )(p0 ) > {(Ref )(gW) : g ∈ G}] ⊂ E. Denote the set of all such f ’s by Sep(p0 , G). For each f ∈ Sep(p0 , G), denote [(Ref )(p0 ) > {(Ref )(g ) : g ∈ G}] by Ef . Since Ef ⊂ E, we can choose a representative Ef0 of Ef such that Ef0 ⊂ E. Denote {Ef0 : f ∈ Sep(p0 , G)} by B , then Remark 2.1 shows there exists a sequence {fn : n ∈ N } in Sep(p0 , G) such that If µ(E \

S

n ≥1

Ef0n = esssup(B ). We will show that µ(E 0 \

S

n≥1

Ef0n ) = 0 as follows.

S ) > 0, then by Lemma 3.3 there must be an fH ∈ S for H = E 0 \ n≥1 Ef0n such that fH ∈ Sep(p0 , G) and Ef0H ⊂ H with µ(Ef0H ) ≥ µ(L) > 0, where L is the set associated with H and fH . This contradicts the fact that S 0 n≥1 Efn = esssup(B ). S Now, assuming, without loss of generality, that n≥1 Ef0n = E 0 , we can construct an f ∈ Sep(p0 , G) such that Ef = E, 0

0 n ≥1 E f n

∗

S

which will end the proof of Theorem 3.1. Sn−1 P ˜ Let An = Ef0n \ k=1 Ef0k for each n ≥ 2, and A1 = Ef01 . Define f : S → L(µ, K ) by f (p) = n≥1 IAn · fn (p), ∀p ∈ S. Then Lemma 3.1 shows that f ∈ S ∗ . We can suppose µ(An ) > 0 for each n ≥ 1 (otherwise such an An is automatically removed, and the corresponding term S in the definition of f will automatically disappear!). Then one can easily see that Ef = n≥1 A˜ n = e E0 = E.  The following counterexample will show that the M-convexity of G cannot be weakened to the usual convexity. Example 3.1. Let (Ω , A, µ) be a σ -finite positive measure space without any µ-atom, (B, k · k) a nontrivial normed space over K , L(µ, B) the RN module of µ-equivalence classes of B-valued µ-measurable functions on Ω (see [9, Example 2.1] for details). Take G to be a closed convex subset of B such that G contains at least two different elements and let L(µ, G) denote the closed M-convex subset of L(µ, B) which consists of all the µ-equivalence class of G-valued µ-measurable functions on Ω . We regard G as a closed convex subset of L(µ, B) with each element of G identified with the µ-equivalence classes of the constant function with value equal to this element. W Take an element p0 in L(µ, G) \ G, then it is impossible ∗ that there exists an f ∈ ( L (µ, B )) such that ( Ref )( p ) > {(Ref )(g ) : g ∈ G} since one can easily verify that 0 W W {(Ref )(g ) : g ∈ G} = {(Ref )(g ) : g ∈ L(µ, G)} whenever f ∈ (L(µ, B))∗ . To give another natural application of Theorem 3.1, we first look at an important example of random locally convex modules. Example 3.2. Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ), and S ∗ the random conjugate f space of S. For a given f ∈ S ∗ , define Xf : S → L+ (µ) by Xp = |f (p)|, ∀p ∈ S, and denote {Xf : f ∈ S ∗ } by σ (S , S ∗ ). Then ∗ (S , σ (S , S )) becomes a random locally convex module, the σ (S , S ∗ )-topology is called the random weak topology for S. Dually, one can also consider the random weak star topology for S ∗ , namely σ (S ∗ , S )-topology. Corollary 3.4. Let (S , {Xd }d∈D ) be a random locally convex module, and G an M-convex subset of S. Then G is closed iff it is also σ (S , S ∗ )-closed. Proof. The proof is completely similar to that of the corresponding classical prototype, so is omitted.



Corollaries 3.5 and 3.6 below were first derived from the algebraic form of the Hahn–Banach theorem for random linear functionals in [11]. In the following we will prove that they can also be obtained immediately from Corollary 3.3, see [11] for the relative terminologies in Corollaries 3.5 and 3.6. Corollary 3.5 ([11, Lemma 3.1]). Let S be a topological module over the topological algebra L(µ, K ), X : S → L+ (µ) a continuous random seminormVwith the property that Xϕ·p = ϕ · Xp , ∀ϕ ∈ L+ (µ) and p ∈ S, M a linear subspace of S, and p0 an element in S such that ξ =: {Xp0 −g : g ∈ M } 6= 0. Then there exists a continuous module homomorphism f from S to L(µ, K ) such that the following two conditions are satisfied: (1) f (p) = 0, ∀p ∈ M; (2) f (p0 ) = I[ξ 6=0] . Proof. Since X is a random seminorm with the property that Xϕ·p = ϕ · Xp , ∀ϕ ∈ L+ (µ) and p ∈ S, one can easily verify

Pn

˜ · Xpi for any n disjoint A-measurable sets {Ai : 1 ≤ i ≤ n} and Pn any n elements {pi : 1 ≤ i ≤ n}, where n is a positive integer. Further, for a given simple element ϕ = α · ˜I in Pn i=1 i Ai ˜ L(µ, K ) with {αi : 1 ≤ i ≤ n} ⊂ K and {Ai : 1 ≤ i ≤ n} as above, we have that Xϕ·p = XPn ˜IA ·(αi p) = I i=1 Ai · Xαi ·p = i=1 i Pn ( i=1 |αi | · ˜IAi ) · Xp = |ϕ| · Xp , ∀p ∈ S, so the continuity of X means that Xϕ·p = |ϕ| · Xp , ∀ϕ ∈ L(µ, K ) and p ∈ S. Pn Denote the original topology for S by T . Let L = { i=1 ˜IAi · pi : n ∈ N , {Ai : 1 ≤ i ≤ n} are n disjoint Pn A-measurable sets such that by i=1 Ai = Ω , and pi ∈ M , 1 ≤ i ≤ n}, and G = the T -closed submodule generated V M. Since X is T -continuous and M is a linear subspace, G is the T -closure of L, which certainly implies that {Xp0 −g : that X also has the property that XPn ˜IA ·pi = i=1 i

i=1 IAi

3804

T.X. Guo et al. / Nonlinear Analysis 71 (2009) 3794–3804

Pn V Pn ˜ ˜ for each g = g i=1 IAi · Xp0 −pV i= 1 IAi · pi in L, we have i V ∈ G} = {Xp0 −gV: g ∈ L}. From the fact that Xp0 −gV= V {Xp0 −g : g ∈ L} = {Xp0 −g : g ∈ M }. Thus we have that {Xp0 −g : g ∈ G} = {Xp0 −g : g ∈ L} = {Xp0 −g : g ∈ M } = ξ . ˆ ) of S with respect to X, as obtained in the proof of Lemma 3.3. Then (Sˆ , X ˆ ) is an RN Consider the quotient space (Sˆ , X module over K with base (Ω , A, µ), and the canonical quotient mapping J is a continuous module homomorphism from ˆ ) since X is T -continuous. (S , T ) to (V Sˆ , X V Since {Xˆ J (p0 )−J (g ) : g ∈ G} = {Xp0 −g : g ∈ G} = ξ > 0, we have J (p0 ) 6∈ J (G) (where J (G) is the closure of J (G) ˆ ˆ in (S , X)). Since J (G) is a closed submodule, it is also a closed M-balanced and M-convex subset. Applying Corollary 3.3 to ˆ )∗ such that the following two conditions are satisfied: J (p0 ) and J (G) will yield an F ∈ (Sˆ , X W ˆ (3) |F (J (p0 ))| > {|F (ˆg )| : g ∈ J (G)}; W (4) [|F (J (p0 ))| > {|F (ˆg )| : gˆ ∈ J (G)}] = [ξ 6= 0]. W W Since J (G) is a submodule and F is continuous, (3) implies that {|F (J (g ))| : g ∈ G} = {|F (ˆg )| : gˆ ∈ J (G)} = 0. Further (4) implies that [|F (J (p0 ))| > 0] = [ξ 6= 0]. Define f : S → L(µ, K ) by f (p) = Q ((F ◦ J )(p0 )) · (F ◦ J )(p), ∀p ∈ S. Then f is a continuous module homomorphism from (S , T ) to L(µ, K ) such that f (p0 ) = I[(F ◦J )(p0 )6=0] = I[|(F ◦J )(p0 )|>0] = I[ξ 6=0] . Clearly f (g ) = 0, ∀g ∈ G, which in turn implies f (p) = 0, ∀p ∈ M.  Corollary 3.6 ([11, Corollary 3.1]). Let (S , {Xd }d∈D ) be a random locally convex module over K with base (Ω , A, µ), M a closed submodule of S, and p0 ∈ S \ M. Then there exists a continuous module homomorphism f from S to L(µ, K ) such that f (p) = 0, ∀p ∈ M, and f (p0 ) 6= 0. Proof. Since it is obvious that M is a closed from Corollary 3.3 that there W M-balanced and M-convex subset of S, it follows W exists an f ∈ S ∗ such that |f (p0 )| > {|f (g )| : g ∈ M }, which in turn also implies {|f (g )| : g ∈ M } = 0, so f is as desired.  Remark 3.4. With the deep development of the theory of random conjugate spaces, the theory of random duality should naturally be studied. Practically, we have studied some compatible invariants with respect to random duality, and Theorem 3.1 together with the corollaries to it has played an essential role in establishing compatible invariants with respect to the random setting-random duality. Acknowledgement The authors would like to thank the kind referee for a large number of invaluable comments which considerably improved the readability of this paper. References [1] N. Dunford, J.T. Schwartz, Linear Operators, Interscience, London, 1957. [2] R.R. Phelps, Convex Functions, Differentiability and Monotone Operators, in: Lecture Notes in Mathematics, vol. 364, Springer-Verlag, Berlin, 1989. [3] T.X. Guo, The Radon–Nikody´ m property of conjugate spaces and the w ∗ -equivalence theorem on the w ∗ -measurable functions, Sci. China Ser. A 39 (1996) 1034–1041. [4] T.X. Guo, A characterization for a complete random normed module to be random reflexive, J. Xiamen Univ. Natur. Sci. 36 (1997) 499–502. [5] T.X. Guo, Some basic theories of random normed linear spaces and random inner product spaces, Acta Anal. Funct. Appl. 1 (1999) 160–184. [6] T.X. Guo, Representation theorems of the dual of Lebesgue–Bochner function spaces, Sci. China Ser. A 43 (2000) 234–243. [7] T.X. Guo, Survey of recent developments of random metric theory and its applications in China (I), Acta Anal. Funct. Appl. 3 (2001) 129–158. [8] T.X. Guo, Survey of recent developments of random metric theory and its applications in China (II), Acta Anal. Funct. Appl. 3 (2001) 208–230. [9] T.X. Guo, S.B. Li, The James theorem in complete random normed modules, J Math. Anal. Appl. 308 (2005) 257–265. [10] T.X. Guo, The relation of Banach–Alaoglu theorem and Banach–Bourbaki–Kakutani–Šmulian theorem in the complete random normed modules to stratification structure, Sci. China Ser. A 51 (2008) 1651–1663. [11] T.X. Guo, S.L. Peng, A characterization for an L(µ, K )-topological module to admit enough canonical module homomorphisms, J Math. Anal. Appl. 263 (2001) 580–599. [12] T.X. Guo, L.H. Zhu, A characterization of continuous module homomorphisms on random seminormed modules and its applications, Acta Math. Sin. (Engl. Ser.) 19 (2003) 201–208. [13] T.X. Guo, Z.Y. You, A note on pointwise best approximation, J Approx. Theory 93 (1998) 344–347. [14] T.X. Guo, Several applications of the theory of random conjugate spaces to measurability problems, Sci. China Ser. A 50 (2007) 737–747. [15] F.Z. Gong, Probabilistic metric spaces with applications to stochastic analysis, Master’s Thesis, Northwestern University, China, 1989. [16] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Dover Publications, New York, 2005. [17] T.X. Guo, X.L. Zeng, Existence of nontrivial continuous linear functionals on random normed modules, Chinese J. Engrg. Math. 25 (2008) 117–123. [18] T.X. Guo, H.X. Xiao, A separation theorem in random normed modules, J. Xiamen Univ. Natur. Sci. 42 (2003) 270–274.