CONVERGENCE OF SET VALUED SUB- AND SUPERMARTINGALES IN THE KURATOWSKI MOSCO SENSE 1. By Shoumei Li and Yukio Ogura Saga University

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1 The nnals of Probability 1998, Vol. 26, No. 3, CONVERGENCE OF SET VLUED SUB- ND SUPERMRTINGLES IN THE KURTOWSKI MOSCO SENSE 1 By Shoumei Li and Yukio Ogura Saga University The purpose of this paper is to prove some convergence theorems of closed and convex set valued sub- and supermartingales in the Kuratowski Mosco sense. To get submartingale convergence theorems, we give sufficient conditions for the Kudo umann integral and Hiai Umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables. We also give an example of a bounded closed convex set valued random variable whose Kudo umann integral is not closed. 1. Introduction. Since umann introduced the concepts of integrals of set valued random variables in 1965 [1], the study of set valued random variables has been developed extensively, with applications to economics and optimal control problems and so on, by many authors; see [3], [9], [21] [23] and so on. In particular, Hiai and Umegaki in [5], and Hiai in [6], [7] presented the theory of set valued conditional expectations and set valued martingales. This theory is the basic foundation of the study of set valued martingale theory and applications. They obtained existence and convergence theorems of conditional expectations, strong law of large number theorems for set valued random variables and the representation theorem of closed set valued martingales. However, there are few results on the closedness of the umann integral of a closed set random variable and the associated conditional expectation. In our previous papers [13] [15], we studied fuzzy valued random variables, conditional expectations and fuzzy valued martingales extensively, based on the Hiai and Umegaki results. mong the results in [13], we used the method of martingale selections especially to obtain a regularity theorem (cf. [13], Lemma 5.7) for closed convex set valued martingales and applied it to fuzzy valued martingales (cf. [13], Theorem 5.1) without using the condition for the dual space to be separable. The regularity of -valued martingales, as we know, implies convergence almost everywhere. However, in the case of closed convex set valued martingales, a regularity property does not imply convergence in the Hausdorff distance (cf. [14], Example 4.2). Thus in [14], we made use of Kuratowski Mosco topology in place of Hausdorff distance, which was a main tool in [13] (and in most previous works of other authors) and got convergence theorems both for closed convex set valued martingales and fuzzy Received pril 1997; revised October Supported by Grant-in-id for Scientific Research MS 1991 subject classifications. Primary 60G42, 28B20; secondary 60G48, 60D05. Key words and phrases. Set valued submartingale, set valued supermartingale, Kudo umann integral, Kuratowski Mosco convergence. 1384

2 SET VLUED SMRTINGLES 1385 valued martingales (cf. [14], Theorems 3.1 and 3.3). There is a rich history on Kuratowski Mosco topology after the celebrated paper [17] (see [18], [24], [25], e.g.). ctually, both the notions of the Hausdorff distance convergence and the Kuratowski Mosco convergence for set valued random variables in a metric space are eminently useful in several areas of mathematics and applications such as optimization and control, stochastic and integral geometry and mathematical economics. However, in a normed space, especially for infinite dimensional cases, the Kuratowski Mosco convergence is more tractable than the Hausdorff. We note that Papageorgiou [19, 20] obtained fruitful results in convergence theorems for set valued random variables as well as for closed convex set valued martingales. However, the assumption there that the conjugate functions are uniformly equi-continuous is not easy to check. In [14], we succeed in dropping that assumption by exploiting martingale selections. This paper is the continuation of the work of [13] [15]. Our main purpose is to get convergence theorems for closed convex set valued submartingales and supermartingales in the Kuratowski Mosco sense. For this purpose, we have to give sufficient conditions for the umann integral and Hiai Umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables. Note that they are not closed in general (cf. the counterexample in Section 2). In Section 2, we give our main results after introducing necessary definitions and notations. Sections 3 to 6 contain the proofs of our main results. It is not hard to extend the convergence theorems here to those for the fuzzy valued sub- and supermartingales as we did in [14]. Here, however, we focus only on the set valued case, because we want to make this paper rather theoretical. 2. Definitions and results. Throughout this paper, ( µ) is a complete probability space. Denote by a real separable Banach space, the dual space of, K the set of all nonempty, closed subsets of, K c the set of all nonempty closed and convex subsets of and K cc the set of all nonempty compact and convex subsets of. For K, welet K = sup x x. The Hausdorff distance on K is defined as follows: 21 { d H B =max sup a inf a b sup inf a b b B a for B K. But if B are unbounded, d H B may be infinite. It is well known (cf. [12], pages 214 and 407) that the family of all bounded elements in K is a complete metric space with respect to the Hausdorff metric d H, and the family of all bounded elements in K c, K cc are closed subsets of this complete space. set valued random variable F K is a measurable mapping; that is, for every B K, F 1 B =ω ; Fω B (cf. [5]). measurable mapping f is called a measurable selection of F if fω Fω for all ω. b B

3 1386 S. LI ND Y. OGUR Denote by L 1 the Banach space of all measurable mappings g such that the norm g L = gω dµ is finite. For a measurable set valued random variable F, define the set S F = { f L 1 fω Fω a.e.µ It then follows that S F is closed in L 1. For a sub-σ-field 0, denote by S F 0 the set of all 0 -measurable mappings in S F. set valued random variable F K is called integrably bounded iff the real valued random variable Fω K is integrable. Let L 1 µk denote the space of all integrably bounded set valued random variables where two set valued random variables F 1, F 2 L 1 µk are considered to be identical if F 1 ω = F 2 ω, a.e.µ. The spaces L 1 µk c, L 1 0 µk, L 1 0 µk c are defined in a similar way. Then F L 1 µk iff there exists a sequence f n of measurable functions f n K such that Fω =clf n ω for all ω. Define the subset of L 1 µk as follows: L 1 µb = { F L 1 µk Fω B aeµ and if 0 is a sub-σ-field of, we denote L 1 0 µb = { F L 1 µb F is 0 -measurable where B is a subset of K. For each F L 1 µk, the Kudo umann integral of F is 22 { Fdµ= fdµ f S F where fdµ is the usual Bochner integral. Define Fdµ = { fdµ f S F, for. Remark 1. The integral Fdµ of F is not closed in general, as is seen from the following counterexample. Example. Let be a nonreflexive separable Banach space. Then there exists a pair of disjoint bounded closed convex sets which cannot be separated by a hyperplane; that is, there are B 1, B 2 K c, such that B 1 B 2 = and d B 1 B 2 =inf x1 B 1 x 2 B 2 d x 1 x 2 =0, where d is the metric derived by (cf. [8]). Let such that µ = 1 2. Define a set valued random variable { B1 if ω Fω = B 2 if ω c Then { Fdµ= fdµ+ fdµ f S F c = { 1 2 x 1 x 2 x 1 B 1 x 2 B 2

4 SET VLUED SMRTINGLES 1387 Since B 1 B 2 =, we then have 0 / Fdµ, but d B 1 B 2 =0 implies 0 cl Fdµ. Let 0 be a sub-σ-field of. The conditional expectation EF 0 of an F L 1 µ K c is determined as an 0 -measurable element of L 1 µk c such that 23 S EF0 0 =cl { Ef 0 f S F where the closure is taken in the L 1 (cf. [5]). If is separable, this is equivalent to the formula 24 cl Fdµ= cl EF 0 dµ for 0 Using the above counterexample, we can easily get a counterexample to explain that the set Ef 0 f S F is not necessarily closed. In the following, we will give some sufficient conditions for the closedness of the Kudo umann integral of a compact convex set valued random variable and of a closed convex set valued random variable F and the set Ef 0 f S F concerning the conditional expectations EF 0 of F. They are important for our proof of the set valued submartingale convergence theorem. Banach space is said to have the Radon Nikodym property (RNP) with respect to a finite measure space µ if, for each µ-continuous -valued measure m: of bounded variation, there exists an integrable mapping f: such that m = fdµ for all. It is known that every separable dual space and every reflexive space has the RNP (cf. [2] and [25]). Theorem 1. (i) If has the RNP and F L 1 µk cc, then the set { Fdµ= fdµ f S F is closed in. (ii) If has the RNP, F L 1 µk cc and 0 is countably generated, that is, 0 = σ for a countable subclass of, then the set Ef 0 f S F is closed in L 1 Theorem 2. Then the set (i) If is a reflexive Banach space, F L 1 µ K c. { Fdµ= fdµ f S F is closed. (ii) If is a reflexive Banach space, F L 1 µ K c and 0 = σ is countably generated. Then the set { Ef0 f S F is closed in L 1

5 1388 S. LI ND Y. OGUR In the following we begin to discuss set valued martingales. Let n n be a family of complete sub-σ-fields of such that n1 n2 if n 1 n 2, and be the σ-field generated by n=1 n. system F n n n is called a set valued martingale iff (1) X n L 1 n µ K c, n, and (2) X n = EX n+1 n, n, a.e.µ. sequence of set valued random variables F n n 1 is called uniformly integrable iff sup F n ω K dµ = 0 λ n F n ω K >λ Chatterji in [2] proved that a Banach space has the RNP with respect to µ iff every uniformly integrable -valued martingale is regular. The -valued martingales with regularity, as we know, imply convergence almost everywhere with respect to µ. In [5], Hiai and Umegaki extended some results of them to set valued martingales. They proved that if a separable Banach space has the RNP and its dual space is also separable, then every uniformly integrably set valued martingale in L µk c is regular. In [13], we used the method of martingale selections to prove the regularity of martingales (cf. [13], Lemma 5.7) and then extended it to fuzzy valued martingales without using the separability of the dual space (cf. [13], Theorem 5.1). But in the case of set valued martingales, regularity property does not imply convergence in the Hausdorff distance (cf. [14], Example 4.2). Thus in [14], we used the Kuratowski Mosco convergence and proved that regularity property implies convergence in the Kuratowski Mosco sense. In this paper, we will continue to work to get sub- and supermartingale convergence theorems. Let B n n be a sequence of closed sets of. We will say that B n converges to B in the Kuratowski Mosco sense [17, 25] (denoted by B n K M B) iff w- sup B n = B = s- inf B n where and w- sup B n = { x = w- x m x m B m m M s- inf B n =x = s- x n x n B n n Remark 2. (i) Since we have s- inf B n s- sup B n automatically, B n K M B iff w--sup B n B -inf B n. (ii) The notions s- inf B n and w- sup B n are different from the set-theoretic notations of inf and sup of a sequence of sets B n n, which we denote by LiB n and LsB n, respectively; that is, LiB n = B n k=1 n k

6 and SET VLUED SMRTINGLES 1389 LsB n = k=1 n k B n The connections between s- inf B n, w- sup B n and LiB n, LsB n are clarified by the following relations: and LiB n s inf B n = Li ( clk 1 B n ) = k=1 k=1 n=1 m n clk 1 B n LsB n w- sup B n where cl (resp. w-cl ) is the closure (resp. weak closure) of the set K and ε is an open ε-neighborhood of the set defined as follows: ε = { x d 1 x <ε where d 1 x =inf x y x (cf. [25]). sequence of set valued random variables F n L 1 µ K c converges to F L 1 µ K c in the Kuratowski Mosco sense (denoted by F n K M F a.e.µ) iff F n ω converges to Fω for almost every ω with respect to µ. In [14], we got the following convergence theorem for nonempty closed and convex set valued martingales in the Kuratowski Mosco sense. Theorem 3. ssume that is a Banach space satisfying the RNP with the separable dual. Then, for every uniformly integrable set valued martingale F n n n, there exists a unique F L 1 µk c such that F n n n is a martingale and F n K M F, a.e.µ. system F n n n is called a set valued submartingale iff it satisfies the following two conditions. 1. For each n, F n L 1 n µ K c. 2. For each n, S Fn n Ef n f S Fn+1 n+1 Remark 3. (i) Condition 2 is equivalent to the following. 3. For each n m with n m, S Fn n Ef n f S Fm m (ii) Condition 2 is stronger than the notion of submartingale in [5], where they use the condition that is, F n ω EF n+1 n ω aeµ 2. For each n, S Fn n clef n f S Fn+1 n+1.

7 1390 S. LI ND Y. OGUR (iii) Condition 2 is equivalent to condition 2 if Ŝ = { Ef n f S Fn+1 n+1 is closed. Concerning the closedness of it, we have given sufficient conditions in Theorems 1 and 2. Now we give a convergence theorem for set valued submartingales as follows. Theorem 4. ssume that is a Banach space satisfying the RNP with the separable dual. Then, for every uniformly integrable set valued submartingale F n n n, there exists a unique F L 1 µk c such that F n K M F, a.e.µ. system F n n n is called a set valued supermartingale iff it satisfies the following two conditions. 1. For each n, F n L 1 n µ K c. 2. For each n, EF n+1 n ω F n ω a.e.µ Remark 4. Condition (2) is equivalent to EF m n ω F n ω a.e.µ for each n, m with n m. The following is the convergence theorem for set valued supermartingales in the Kuratowski Mosco sense. Theorem 5. ssume that is a Banach space satisfying the RNP with the separable dual, F n n n is a uniformly integrable set valued supermartingale and 25 M = { f L 1 µ Ef n S Fn n n=1 is a nonempty set. Then there exists a unique F L 1 µ K c such that F n K M F, a.e.µ. 3. Proof of Theorem 1. In this section, we will prove Theorem 1. For our proof, we need the following lemmas. Lemma 3.1 [5]. Let F L 1 µk; then there exists a sequence f n S F such that Fω =clf n ω for all ω. Lemma 3.2 [5]. Let F L 1 µk cc. Then f S F iff fdµ cl Fdµ for all. Moreover if is separable, then the same is true for any F L 1 µk c.

8 SET VLUED SMRTINGLES 1391 ProofofTheorem 1(i). Step 1. Taking an F L 1 µk cc, we prove that there exists a countable field such that F is 1 -measurable, where 1 = σ. Indeed, since F L 1 µk cc, by virtue of Lemma 3.1, there exists a sequence f n S F such that Fω =clf n ω for all ω. For every f n, there exists a sequence of simple functions f m n such that f n f m L 1 = 0 n Let m = { f m n n 1 x 0 x 0 f m mn We then have that is countable. Thus the field, generated by, is also countable (cf. [4], Lemma 8.4, page 167). Hence F is 1 -measurable, where 1 = σ =σ. Step 2. We prove that cl Fdµ= cl { n fdµ f S F is a compact set in for any. Indeed, if F is a simple set valued random variable, that is, if Fω = n K i I i ω i=1 K i K cc then { n cl Fdµ= cl x i µ i x i K i i=1 ( n ) = cl K i µ i K cc i=1 If F L 1 µk cc, then there exists a simple set valued random variable F n such that d H F n ωfω 0, a.e.µ. By [13], Lemma 2.2, we have ) ( d H (cl F n dµ cl Fdµ d H Fn ωfω ) dµ Thus d H cl F n dµ cl Fdµ 0. Then cl Fdµis compact since K cc is closed with respect to the Hausdorff metric. Step 3. For each x cl Fdµ, there exists a sequence f n S F such that x f n dµ = 0

9 1392 S. LI ND Y. OGUR Due to the inclusion f n dµ n cl Fdµ K cc, for any, there exists a subsequence g n of f n such that λ = g n dµ Since is countable, with the help of the Cantor diagonal procedure, there exists a subsequence, also denoted by g n, such that λ = g n dµ for each In view of g nω dµ Fω K dµ, for each, we see that λ n = g nω dµ n is uniformly countably additive on 1, that is, for any disjoint sequence i 1, sup λ n i 0 as N n i=n Thus there exists a subsequence, which we also denote as g n, such that λ = g n dµ for each 1 (cf. [4], Lemma 8.8, page 292). Step 4. We first note the following. (a) λ is µ-continuous, that is, µ n 0 implies λ n 0, where n 1. This is due to [4], Theorem 7.2, page 158. (b) λ is of bounded variation, that is, for each 1 n sup λ i < i i=1 where i i = 1n is any measurable finite partition of. (c) λ is countably additive, which is due to [4], Corollary 7.4, page 160. Since X has the RNP, these ensure the existence of a g L 1 such that λ = gdµ for each 1 Further, we have gdµ= g n dµ cl F dµ 1 Hence, by Lemma 3.2, we have g S F. Especially, let = to obtain gdµ= g n dµ = x Thus we get { x Fdµ= fdµ f S F

10 SET VLUED SMRTINGLES 1393 ProofofTheorem 1(ii). Step 1. Let g clef 0 f S F. Then there exists a sequence f n S F such that 31 g Ef n 0 dµ = 0 Using the same method as Step 1 in part (i), we can get a countable field 0 such that F is 1 -measurable, where 1 = σ 0. Let be the smallest field including and 0. Then is countable. Furthermore, since { f n dµ cl Fdµ K cc using the Cantor diagonal procedure, we can choose a subsequence g n of f n such that λ = g n dµ for each In the same way as in Step 3 in (i), we have a subsequence, also denoted by g n, such that λ = g n dµ for each 0 1 Following Step 4 of (i), we then get an f L µ such that λ = f dµ 0 1 In view of 32 fdµ= g n dµ cl F dµ 0 1 and Lemma 3.2, we have f S F 0 1. Step 2. Noting that g φ g L 1φ L g L 1 φ L and (3.1), we have Ef n 0 φdµ = g φ dµ φ L For any 0 and x, let φω =I ωx L. We then have Ef n 0 dµ x = g dµ x and combining this with Ef n 0 dµ = f n dµ

11 1394 S. LI ND Y. OGUR we get f n dµ x = g dµ x x 0 This with the relation g n f n implies g n dµ x = Combining this with (3.2), we obtain fdµ= g dµ g dµ x x 0 0 Since g is 0 -measurable, we have g = Ef 0. That is, g Ef 0 f S F. 4. Proof of Theorem 2. ProofofTheorem 2(i). Step 1. In the same way as in the proof of Theorem 1, we can find a countable field such that F is σ -measurable. Step 2. For each x cl Fdµ, there exists f n S F such that x f n dµ = 0 Fix an. Since is reflexive and f n dµ n is bounded, we have that f n dµ n is weak sequentially relatively compact (cf. [5], Theorem 3.28, page 68). By using the Cantor diagonal procedure, we can choose a subsequence g n f n such that 41 λ =w- g n dµ for each For each fixed x, λ n x n is uniformly countably additive on 1, where λ = g n dµ. Thus there exists a subsequence, also denoted by g n, such that λ =w- g n dµ for each 1 Furthermore, we have λ n x x Fω K dµ 1 Indeed, g n dµ x = g n x dµ g n x dµ x Fω K dµ

12 SET VLUED SMRTINGLES 1395 Step 3. For every given x, λ x is µ-continuous, of bounded variation and countable additive. Thus there exists an fx L 1 1 µ such that 42 λx = fx ω dµ 1 Moreover, fax + by =afx +bfy aeµ ab x y and 43 fx ω dµ 2x ( Fω K 1 ) aeµ where a b = maxa b. Indeed, let + = ω fx ω 0 and = ω fx ω 0. Then + 1 and fx dµ = fx dµ + fx dµ = λ + x λ x 2x Fω K dµ 1 Now, for each ε>0, let N =ω fx ω 2 + εx Fω K 1. Then we have 2 + εx ( FωK 1 ) dµ fx ω dµ N N 2x ( FωK 1 ) dµ Thus N Fω K 1 dµ = 0, which implies µn =0. We then have fx ω 2 + εx ( Fω K 1 ) aeµ Letting ε 0, we get (4.3). Let be a countable dense subset of. Then we have 44 fx ω 2x ( Fω K 1 ) x aeµ Since fx is a linear function, fx is uniformly continuous on. Thus we can extend fx to a function on X satisfying (4.2) and (4.4), which we denote by fx again. From (4.4) for all x, there exists an fω such that N fx ω = fωx x and fω 2 ( Fω K 1 )

13 1396 S. LI ND Y. OGUR which implies f L 1 1 µ. Combining this with (4.2), we obtain λx = fωx dµ = f dµ x 1 We then have λ = fdµ for all 1. Step 4. In this step we will show that 45 λ = fdµ cl F dµ 1 Since g n S F,wehave g n dµ x sup x x x x Fdµ Letting n,weget λx sup x x x sup Fdµ x cl x x x Fdµ Thus λ cl Fdµ, and by Lemma 3.2, we have f S F. Step 5. Let = in (4.1). Then we have λ = Since f S F,wehavex Fdµ. fdµ= w- g n dµ = x ProofofTheorem 2(ii). Step 1. Let f n S F, g L 1 0 µ,, 0 and 1 be the same as those in Step 1 in the proof of Theorem 1(ii). By using the same method as in Step 2 in the proof of part (i), the set f n dµ is weakly sequentially relatively compact for each. By the discussion of Steps 2 and 3 in part (i), there exist an f S F 0 1, and a subsequence g n f n, such that fdµ= w- g n dµ 0 1 Step 2. By the same method as in Step 2 in the proof of Theorem 1(ii), we get the conclusion. 5. Proof of Theorem 4. To prove Theorem 4, we need the following lemmas. Lemma 5.1. Let F n n n be a set valued submartingale. Then there exists a system of -valued random variables f k m n kmn such that the following hold. (i) For each km, the sequence f k m n n n is a martingale. (ii) For each k, m, n with 1 m n, f k m n S F n n.

14 SET VLUED SMRTINGLES 1397 (iii) For each n, F n ω =cl { f k m nω 1 m n k for almost every ω Proof. Fix an m. Since F m L 1 µk c, we can find a countable set f k m m k S F m such that F m ω =cl { f k m m ω k ω by virtue of Lemma 3.1. We then choose f k m m+1 S F m+1 m+1 so that Ef k m m+1 m = f k m m. Repeating this procedure, one gets a sequence f k m n n m such that f k m n S F n n Ef k m n+1 n=f k m n n m Defining f k m n = Efk m m n for 1 n<m, we get the system of martingales f k m n n n, km which also satisfy conditions (ii) and (iii). Lemma 5.2 [5]. Let M be a nonempty bounded closed convex subset of L. Then there exists an F L 1 µk c such that M = S F iff M is decomposable, that is, iff h = fi + gi \ belongs to M for all and f g M. Lemma 5.3 [14]. Let f k n n k be a sequence of real valued submartingales such that sup n Esup k f k n + <, and suppose that It then holds that fk n = fk sup f k n = sup f k k k aeµ k aeµ ProofofTheorem 4. Step 1. Let [ { 51 S = cl f L 1 µ Ef n S Fn n ] m=1 n=m Clearly S is a closed convex bounded set. It is nonempty. Indeed, since the set valued submartingale F n n n is uniformly integrable, each -valued martingale f k m n n in Lemma 5.1 is uniformly integrable. Thus there exists an f k m L µ such that f k m n fk m, a.e.µ. Further it converges in L 1 µ, so that f k m S. Finally, S is decomposable. Indeed, for each f g S, we can find two sequences f j and g j such that f j f, g j g and f j, g j Sm j, where Sm = { f L 1 µ Ef n S Fn n n=m

15 1398 S. LI ND Y. OGUR Then h j = I f j + I \ g j Sm j by the same reason as in the proof of [13], Lemma 5.7. Now, by virtue of Lemma 5.2, there exists a unique F L 1 µ K c such that S = S F Step 2. We will show that 52 F s- inf F n aeµ Take an f S F = S; then there is a sequence f j and a subsequence m 1 <m 2 < <m j <m j+1 < of such that 53 f f j L 1 2 j f j Sm j j Thus we have [ E f j ω fω ] = E [ f j ω fω ] j=1 = j=1 j=1 1 2 j < This implies j=1 f j ω fω <, a.e.µ. Moreover, we have f j ω fω 0 as j aeµ Now noting that f j Sm j, we have Ef j n S Fn n ; that is, Ef j n ω F n ω a.e.µ, for each n m j, and f j ω Ef j n ω 0 as n aeµ Since F n ω is a closed set in, this implies f j ω s- inf F n ω. But the sets s- inf F n ω are closed for a.e.ω w.r.t µ. Hence, we get fω s- inf F nω aeµ Step 3. We will show that 54 sup x x x F n ω sup x x x F ω x aeµ Taking the system f k m n kmn in Lemma 5.1, we have 55 sup x x= sup x f k m n ω ω n x F n ω k 1 m n Since f k m n n is a -valued uniformly integrable martingale, there exists an f k m Sm S F such that fk m n = fk m and fk m n = E( f k m ) n aeµ

16 SET VLUED SMRTINGLES 1399 which implies x f k m n =x f k m aeµ for each x and k. Furthermore, the sequence x f k m n n n is a real valued martingale with [ ] x f k + m n sup E n sup k 1 m n x sup E [ ] F n K < n where a + = maxa 0. Indeed, for m n, k, condition (ii) in Lemma 5.1 ensures f k m n F n K. Now, by virtue of Lemma 5.3, it follows that for every fixed x, sup x f k m n ω = sup x f k m ω aeµ k 1 m n Since is separable, we get km sup k 1 m n x f k m n ω = sup x f k m ω km x aeµ We thus obtain from (5.5) that sup x x= sup x f k m ω x aeµ x F n ω km On the other hand, we have f k m F ω aeµ k, since f k m S F. Thus we get x f k m ω sup x x x F ω x aeµ This implies sup x f k m ω sup x x km x F ω x aeµ ensuring (5.4). Step 4. We now show that 56 w- sup F n F aeµ which together with (5.2) completes the proof of the theorem. Take an x w- sup F n ω. Then, by the definition, there exists a subsequence n i such that x ni F ni, i, x ni x in the weak sense, and This implies i x x ni =x x x x x sup i x F ni ω x x

17 1400 S. LI ND Y. OGUR and together with (5.4), x x sup x x x F ω aeµ We thus obtain x F ω a.e.µ, from the argument in [5], page Proof of Theorem 5. Step 1. Let M be that given in (2.4). We can use the same method as in [13], Lemma 5.7 to get that M is closed, convex, bounded and decomposable. From the assumption of M being nonempty, there exists a unique F L 1 µk c such that S F =M by Lemma 5.2. Now we prove 61 F ω s- inf F nω aeµ Indeed, take an f S F. We then have Ef n S Fn n, that is, Ef n ω F n ω, a.e.µ for all n. Further, since F n is uniformly integrable, it holds that Ef n ω fω = 0 aeµ We thus obtain f S s- inf F n, so that S F S s- inf F n. Step 2. We will prove 62 w- sup F n F aeµ which together with (6.1) completes the proof of the theorem. Let 63 M n = n { f L 1 µ Ef m S Fm m m=1 It is also easy to prove that M n is closed, convex, bounded and decomposable. Since M being nonempty, M n is nonempty for each n. Thus there exists a unique G n L 1 µk c such that S Gn =M n by Lemma 5.2. First, we note that S Fl S Gn, for each l n. Indeed, for each f S Fl l S Fl l, f is l -measurable and fω F l ω, aeµ, so that f is -measurable, and Ef k cleg k g S Fl l = S EFl k k for 1 k n. Since F l l l is supermartingale, S EFl k S Fk k. We then have Ef k S Fk k, for 1 k n. Thus f S Gn. By virtue of [13], Lemma 2.1, we have F l ω G n ω aeµ for each l n Since G n ω are convex and closed sets in for a.e. ω w.r.t. µ, and convexity and closedness imply weakly closedness, we have w- sup F l ω G n ω l aeµ for each n

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