Convex Risk Measures: Lebesgue Property on one Period and Multi Period Risk Measures and Application in Capital Allocation Problem
aa r X i v : . [ q -f i n . R M ] A p r Convex Risk Measures: Lebesgue Property on onePeriod and Multi Period Risk Measures andApplication in Capital Allocation Problem
Hirbod Assa ∗ [email protected] Department of Mathematics and Statistics &Group for Research in Decision Analysis (GERAD)University of Montreal
Abstract
In this work we study the Lebesgue property for convex risk mea-sures on the space of bounded c`adl`ag random processes ( R ∞ ). Lebesgueproperty has been defined for one period convex risk measures in [16]and earlier had been studied in [9] for coherent risk measures. Weintroduce and study the Lebesgue property for convex risk measuresin the multi period framework. We give presentation of all convexrisk measures with Lebesgue property on bounded c`adl`ag processes.To do that we need to have a complete description of compact sets of A . The main mathematical contribution of this paper is the charac-terization of the compact sets of A p (including A ). At the final partof this paper, we will solve the Capital Allocation Problem when wework with coherent risk measures. Keywords :Convex risk measure, Lebesgue property, Bounded c`adl`agProcesses, Capital Allocation Problem
Assessing and qualifying risk is important in many aspects of humanactivities. It seems rational to expect something ”good” by accepting ∗ Pavillon Andr´e-Aisenstadt 2920, chemin de la Tour, bureau 5190 Montreal, QuebecH3T 1J4 omething ”bad” but the question is how much we should be rewarded.Is it ”good enough” to accept the risk? The other question is amongseveral risk exposures how to choose the one that represent a smallerrisk .Markowits ,economist, worked to make explicit the trade-off of therisk and reward in the context of portfolio of financial assets,see [19],[20] and [18]. He has already considered that the returns distributionof assets is jointly Normal (or Gaussian).The variance is a risk measure which assesses both sides of theProfit and Loss distribution. That means variance takes into accountthe losses as well as profits. Regards this problem and other technicalproblems the new measure, Value at Risk, was defined and popularizedin 1994 : V aR α ( X ) = − inf { x | P [ X ≤ x ] > α } Value at Risk (VaR) measures the left tail of P& L random variablewhich is connected with the Losses. In other words VaR is a measureshowing how the market value of an asset or of a portfolio of assets islikely to decrease over a certain time period, under usual conditions.It is typically used by security houses or investment banks to measurethe market risk of their asset portfolios (market value at risk), butis actually a very general concept that has broad application. Asreferences one can consult [8],[13],[15],[16].Unlike variance, VaR does not meet the sub additivity property.Regarding to this problem, in 1998 Artzner et al[2] proposed an ax-iomatic way to define a generation of risk measures called coherentrisk measures. Their idea is to define a risk which can be used ascapital requirements, to regulate the risk assumed by market partici-pants, traders, insurance underwriters, as well as to allocate existingcapital. They propose the following definition [2]:
Definition . A function ρ : L ∞ → R is a Coherent Risk measure if1- ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) for any X, Y ∈ L ∞ and λ ∈ [0 , ρ ( λX ) = λρ ( X ) for any X ∈ L ∞ and λ > ρ ( X + m ) = ρ ( X ) − m for any X ∈ L ∞ and m ∈ R .(TranslationInvariant)4- ρ ( X ) ≤ ρ ( Y ) ∀ X, Y ∈ L ∞ and Y ≤ X .(Decreasing)Condition 1 can be replaced by subadditivity condition, ρ ( X + Y ) ≤ ρ ( X ) + ρ ( Y ). Each condition stated in responds to economic reasons ut they remain somehow controversial. In their seminal work [2]the authors also define the new risk measure, Expected Shortfall, asa particular example of coherent risk measures. Expected Shortfallmeasures the expectation of losses below Value at Risk. The definitionis as follows: ES [ X ] = E [ X | X < V aR α ( X )] . Later F¨ollmer et al [12] proposed to generalize the coherent risk ina natural way to convex risk measure by relaxing the condition 2. Seefor example [12].The main subject of [9] and [12] is the characterization of risk mea-sures with the so-called Fatou property (see next section). ActuallyFatou property is counterpart of the concept of lower semi continuityin weak star topology. The other property which seems essential tostudy the risk measures is the so-called Lebesgue property (see nextsection). This property first time has studied under this name for con-vex risk measures in [16], but earlier this property had been studiedfor coherent risk measures in [9]. The author of [9] showed how thecoherent risk measures with Lebesgue property can solve the CapitalAllocation Problem (see section 4). To study the Lebesgue propertywe need to know the complete description of weak compact sets of L .After some discussion on risk measures for random variables, natu-rally the next step is to define the risk measures for random processes.Artzner et al in [3] gave a simple way to generalize the coherent riskmeasures for discrete time models. For the first time the convex risktheory was defined and studied for continuous time random process in[6]. The authors of [6] defined the coherent and convex Risk measuretheory for bounded c`adl`ag processes. Then they extend the theory ofconvex risk measures on the space of unbounded c`adl`ag processes in[7]. The main subject of [6] is restating and characterizing the Fatouproperty forrisk measures on bounded c`adl`ag processes. Unlike theFatou property, Lebesgue property has not define and studied yet andour main subject in this paper is to define and study Lebesgue prop-erty for bounded c`adl`ag processes. Actually the main mathematicalcontribution of this paper is the characterization of the weak compactsets of A (see section 2.2) which is assumed as the dual space of c`adl`agprocesses. The main application of our result is finding the solutionof Capital Allocation Problem in the c`adl`ag processes framework.The paper is organized as follows: in section 2, within two subsec-tions we give the preliminarily definitions and results for one periodand multi period risk measures. In section 3 we give some definitions nd remarks which are needed for our results in section 4. In section4 we give our main mathematical contributions of this paper. In sec-tion 5 we discuss the Capital Allocation Problem in the frameworkof c`adl`ag processes. Section 6 is devoted to conclusions and finally inappendix we give the proof of theorems 4.2. In this section we briefly review the concepts which preliminarilyshould be studied. In the first subsection we give preliminary defi-nitions and results on one period convex risk measure. In the secondsubsection our subject is to give the same definitions of subsection 1for multi period convex risk measures and we study the same resultson the space of random processes.
We give the following definition from [9] and [16]:
Definition :A convex risk ρ is has Fatou property if for any boundedsequence X n in L ∞ which converges in probability to X then: ρ ( X ) ≤ lim inf ρ ( X n ) . and we say ρ has Lebesgue property if always the equality occurs.Fatou property seems to find its source in the classical theory of locallysolid Riesz spaces with Fatou topologies[1]. Actually the following the-orem shows that Fatou property is defined as the probabilistic coun-terpart of the concept of lower semi continuity in weak star topology[12]. Theorem 2.1 :Let ρ : L ∞ −→ R be a convex risk measure and let P be the set of measures Q such that Q ≪ P . Then the following areequivalent:(i) The function ρ has Fatou property .(ii) There is a penalty function α : P → ( −∞ , + ∞ ] such that: ρ ( X ) = sup Q ∈ P {− E Q [ X ] − α ( Q ) } . (2.1) rom the theory of convex functions we know that one can get α = ρ ∗ where ρ ∗ is the conjugate function: ρ ∗ ( Q ) = sup X { E Q [ X ] − ρ ( X ) } . (2.2)By the theory of convex functions and relation 2.1 we know that forany c ∈ R , the contour set { ρ ≤ c } is convex and weak* closed.On the other hand the contour set { ρ ∗ ≤ c } is always weakly closedsubset of L . The Lebesgue property gives more information aboutthis contour set. Actually we have the following theorem from [16]: Theorem 2.2
Let ρ : L ∞ (Ω) −→ R be a risk measure. The followingconditions are equivalent:1- ρ has Lebesgue property2- { µ ∈ L | ρ ∗ ( µ ) ≤ c } is σ ( L , L ∞ ) -compact subset of L for every c ∈ R + . If we want to restate this theorem for coherent risk measures weshould say ρ has Lebesgue property iff the set { µ | ρ ∗ ( µ ) = 0 } is weaklycompact. Remark . From Dunford-Pettis Theorem we know that a subset of L is relatively weak compact iff it is uniformly integrable. That meanswe could say { µ ∈ L | ρ ∗ ( µ ) ≤ c } is uniformly integrable, instead ofweak compact. In the real world the problems are in the time periods and the fluc-tuation of the financial or economical variables during these periodsare important. To have a measure that can measure the risk of ”ran-dom processes” we should restrict the study on a special spaces ofrandom processes. On the other hand when we work with the randomprocesses the flow of information (i.e. filtration) should be taken intoaccount. In the sequel we assume that all objects are defined in astandard probability space without any atom (Ω , F , P ). This space isendowed with a filtration ( F t ) ≤ t ≤ T satisfying the usual conditions. et R q be as follows: R q = X : [0 , T ] × Ω −→ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X is c`adl`ag X adapted( X ) ∗ ∈ L q , (2.3)This space is equipped with the following norm: k X k R q = k X ∗ k q , (2.4)where X ∗ ( ω ) = sup ≤ t ≤ T | X t ( ω ) | , ω ∈ Ω.In [6], the authors have suggested to use the space of c`adl`ag boundedprocesses as the space of risky financial items. They also suggest touse the space A p as dual. This space is defined as follows: a : [0 , T ] × Ω −→ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = ( a pr , a op ) ,a pr , a op right continuous ,finite variation a pr predictable , a pr = 0 a op optional , purely discontinuousVar( a pr ) + Var( a op ) ∈ L p , (2.5)where Var( f ) is the variation of function f . The space A p isequipped with the following norm: k ( a pr , a op ) k A p = k Var( a pr ) + Var( a op ) k L p . (2.6)The dual relation between A p , R q is defined as: h X, a i = E [( X | a )] , (2.7)where ( X | a ) = T R X t − da pr t + T R X t da op t . Again following [6], let D σ := { a ∈ A ; k a k = 1 } , (2.8)where A = { a = ( a pr , a op ) ∈ A | a pr , a op are non-decreasing } Before moving on with our discussion we need to lay down a fewdefinitions.
Definition
A convex risk measure is a function ρ : R ∞ −→ R inwhich : - ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) for any X, Y ∈ R ∞ and0 ≤ λ ≤ ρ ( X + m ) = ρ ( X ) − m for any X ∈ R ∞ and m ∈ R .(TranslationInvariant)3- ρ ( X ) ≤ ρ ( Y ) ∀ X, Y ∈ R ∞ and Y ≤ X .(Decreasing)We call it coherent if in addition:4- ρ ( λX ) = λρ ( X ) for any X ∈ R ∞ and λ > R ∞ . Definition
The convex function ρ has Fatou property if for any boundedsequence { X n } n ∈ N ⊆ R ∞ , in which for some X ∈ R ∞ , ( X n − X ) ∗ P −→ ρ ( X ) ≤ lim inf ρ ( X n ).From [6], we have the following characterization : Theorem 2.3
Let ρ : R ∞ −→ R be a risk measure. The followingare equivalent:1- ρ is represented as: ρ ( X ) = sup a ∈D σ {−h X, a i − γ ( a ) } , X ∈ R ∞ (2.9) where γ is a so-called penalty function γ : D σ → ( −∞ , + ∞ ] such that −∞ < inf a ∈D σ γ ( a ) < ∞ .2- ρ is a convex risk measure on R ∞ such that { X ∈ R ∞ | ρ ( X ) ≤ } is σ ( R ∞ , A ) -closed.3- ρ has Fatou property.4- ρ is continuous for bounded increasing sequences.Moreover, in each case, the conjugate function ρ ∗ , restricted to D σ , isa penalty function which is bigger than γ and γ can be replaced by ρ ∗ in the first expression above. ater on, in [16] we find the notion of Lebesgue property for riskmeasures on L ∞ (Ω). In this paper we aim to characterize this prop-erty for risk measures on R ∞ and so we need the following extendeddefinition: Definition
The convex function ρ has Lebesgue property if for anybounded sequence { X n } n ∈ N ⊆ R ∞ , in which for some X ∈ R ∞ , ( X n − X ) ∗ P −→
0, we have ρ ( X ) = lim ρ ( X n ).In [16], we find the following characterization theorem for convexrisk measures on L ∞ (Ω): Theorem 2.4
Let ρ : L ∞ (Ω) −→ R be a risk measure. The followingconditions are equivalent:1- ρ has Lebesgue property2- { µ ∈ L | ρ ∗ ( µ ) ≤ c } is σ ( L , L ∞ ) -compact subset of L for every c ∈ R + . One of the main contributions of this paper is to give a character-ization theorem analogous to Theorem 2.4 for risk measures on R ∞ . In this section we give some definitions that are needed in Section 4.Consider ˆ F t = F and ( ˆ R q , ˆ A p ) are the corresponding processspaces. Let Π op and Π pr be the optional and predictable projections.We also show the dual optional and predictable projection of finitevariation processes with the same notation (as references see [6],[11]and [17]). We define the projection Π ∗ : ˆ A p → A p as follows : let a = ( a l , a r ) ∈ ˆ A p . Let ˜ a l = Π pr ( a l ) and ˜ a r = Π op ( a r ). Then one cansplit ˜ a r uniquely to purely discontinuous finite variation part ˜ a rd andcontinuous finite variation part ˜ a rc with ˜ a rc (0) = 0. Now Define:Π ∗ ( a ) = (˜ a l + ˜ a rc , ˜ a rd ) . We know that every predictable process is also optional so ˜ a l , ˜ a rc , ˜ a rd allare optional. This fact by definition of Π ∗ give that for every X ∈ R q we have: h X, a i = h X, Π ∗ ( a ) i . (3.1) or more details see relation 3.5 , Remark 3.6 [6].Let a = ( a pr , a op ) ∈ A p . Since any predictable process is optionalthen by Theorem 2 . .
53 [17] the measure µ ( A ) = h A , a i is optionaland then h X, a i = h Π op ( X ) , a i . This relation with 3.1 give that ∀ X ∈ ˆ R q , a ∈ ˆ A p : h Π op ( X ) , a i = h Π op ( X ) , Π ∗ ( a ) i = h X, Π ∗ ( a ) i . (3.2)For more details reader is referred to [6] Remark 2.1 and Remark 3.6and [17].For every random variable X ∈ L q (Ω , F ) , we identify the constantrandom process X t := X and X . Remark3.1 :Relation 3.1 (or 3.2) shows that Π ∗ is σ ( ˆ A p , ˆ R q )/ σ ( A p , R q )continuous. Remark3.2 :Let X ∈ L q (Ω) be a random variable. By Doob’s Stop-ping Theorem it is easy to see that the optional projection of con-stant random process X is the martingale M t := E [ X |F t ]. So then ∀ X ∈ L q , a ∈ ˆ A p + : E [Var( a ) X ] = h X, a i = h Π op ( X ) , a i , (3.3)Following the paper [6] define ˆ ρ = ρ ◦ Π op . Definition
For every convex function ρ on R q the static convex func-tion due to ρ is defined on L q (Ω , F ) as follows:¯ ρ ( X ) := ˆ ρ ( X ) , ∀ X ∈ L q (Ω , F ) . Remark3.3 :As one can see the static convex functions due to ρ andˆ ρ are the same. On the other hand the arguments in the proof ofTheorem 3.1 [6] shows that if ( X n − X ) ∗ P −→ op X n − Π op X ) ∗ P −→
0. So ρ has Lebesgue property iff ˆ ρ has. Remark3.4
By Theorem 2.1 every coherent risk measure could beidentified with a subset P of D σ . Let A = Var( P ). Then by relation3.3 it is easy to see that: ¯ ρ ( X ) = E A [ − aX ] . (3.4) By the abbreviation r.c. we mean relatively compact. heorem 4.1 Let ρ : R ∞ −→ R be a convex function in which ρ ( − X ) is a convex risk with Fatou property. Then the following areequivalent:1- ρ has Lebesgue property.2- ∀ c ∈ R + the set { a ∈ A | ρ ∗ ( a ) ≤ c } is r.c. for topology σ ( A , R ∞ ) .3- ¯ ρ has Lebesgue property.4- ∀ c ∈ R + the set { f ∈ L | ¯ ρ ∗ ( f ) ≤ c } is r.c. for topology σ ( L , L ∞ ) . Before giving the proof we give the following Theorem which is one ofthe most important contributions of thia paper. The proof is given inthe Appendix.
Theorem 4.2
Let A ⊂ A p and p + q = 1 . The following three con-ditions are equivalent:1- A is r.c. in the topology σ ( A p , R q ) .2-Var ( A ) is r.c. in the topology σ ( L p , L q ) .3- C := { a T − a | a ∈ A } is r.c. in the topology σ ( L p , L q ) . In addition we have two following Corollary.
Corollary 4.3
Let A ⊆ A . Then A is σ ( A , R ∞ ) − r.c. iff Var ( A ) is uniformly integrable. Corollary 4.4
The set A ⊆ A p , for p = ∞ , is σ ( A p , R q ) − r.c. iff itis sequentially r.c. Proof of theorem 4.1 (1) ⇒ (3). It comes out from definition.(3) ⇒ (4). Just Theorem 2.2.(4) ⇒ (2). Let a ∈ A be such that ρ ∗ ( a ) ≤ c for some positive number c . Then by definition of conjugate function ∀ X ∈ R ∞ we have h X, a i− ρ ( X ) ≤ c . Particularly this is true for every random process likeΠ op ( X ) where X ∈ L ∞ . By 3.3 we get E [Var( a ) X ] − ¯ ρ ( X ) ≤ c . Sowe have Var( { a ∈ A | ρ ∗ ( a ) ≤ c } ) ⊆ { µ ∈ L | ¯ ρ ∗ ( µ ) ≤ c } . Thatmeans Var( { a ∈ A | ρ ∗ ( a ) ≤ c } ) is r.c. for topology σ ( L , L ∞ ) and byTheorem 4.2 { a ∈ A | ρ ∗ ( a ) ≤ c } is r.c. for topology σ ( A , R ∞ ). ⇒ (1). First we consider that ρ is positive homogeneous. By thisassumption, for every real number c >
0, the set { a ∈ A | ρ ∗ ( a ) ≤ c } is equal to { a ∈ A | ρ ∗ ( a ) = 0 } =: A .Let X n be a bounded sequence in R ∞ for which for some X ∈ R ∞ ,( X n − X ) ∗ P −→
0. Since ρ is positive homogeneous (then sub additive)and increasing we have : | ρ ( Z ) − ρ ( Y ) | ≤ ρ ( Z − Y ) + + ρ ( Y − Z ) + , ∀ Z, Y ∈ R ∞ . By this relation we could consider X n ≥ X = 0 and ( X n ) ∗ P −→
0. Bythe hypothesis (2) , A is r.c. for topology σ ( A , R ∞ ). So by Lemma4.2 the closed convex set Var( A ) is σ ( L , L ∞ )-compact and as a con-sequence (by Theorem 2.2) the convex function X sup f ∈ Var( A ) E [ f X ]has Lebesgue property. Now by relation 2.9 we have: ρ ( X n ) = sup a ∈ A h X n , a i ≤ sup f ∈ Var( A ) E [( X n ) ∗ f ] n −→ . Let consider the convex function ρ is not necessarily positive homoge-neous. Let X n and X be bounded in R ∞ such that ( X n − X ) ∗ P −→ X n is uniformly bounded then there is a bounded sequence of c n ∈ R + and a positive number ǫ such that: ρ ( X n ) ≤ sup ρ ∗ ( a ) ≤ c n h X n , a i − c n + ǫ. Let c be a cluster point of c n and I ⊆ N such that | c n − c | < ǫ for all n ∈ I .Let ρ ( X ) := sup { ρ ∗ ( a ) ≤ c + ǫ } h X, a i . Since ρ is positively homogeneous ,it has Lebesgue property. Now we have : ρ ( X ) ≥ sup { ρ ∗ ( µ ) ≤ c + ǫ } h Xµ i − c − ǫ = ρ ( X ) − c − ǫ = lim n ∈ I ρ ( X n ) − c − ǫ ≥ lim n ∈ I sup ρ ∗ ( µ ) ≤ c n h X n , µ i − c − ǫ ≥ lim n ∈ I ρ ( X n ) − ǫ ≥ lim inf ρ ( X n ) − ǫ. ince ǫ > (cid:3) In this section we try to give the application of last section in CapitalAllocation Problem with the Fuzzy game approach. This problem forone period coherent risk measures for the first time has been mentionedin earlier work [10]. In [10] the author has defined the weak star subgradient of a coherent risk measure. The author showed that theexistence of solution for the capital allocation problem is equivalentto having nonempty sub gradient. By the main theorem of last sectionwe are now able to show that for coherent risk measures , when thecapital allocation problem has a solution.Here we briefly give the definition of the Capital Allocation Prob-lem , for more details reader is referred to [10],[4],[2] and [5]. Let X , ..., X N be N random process in R ∞ which present N financialitems. The total capital required to face the risk is ρ ( N P i =1 X i ) = k , andwe are to find a ”fair” allocation k , ..., k N so that k + ... + k N = k .An allocation k , ..., k N with k = k + ...k N is called fair if ∀ α j , j =1 , ..., N, ≤ α j ≤ X j α j k j ≤ ρ ( X j αX j ) . Now we give the definition and theorems which we need to find afair allocation.
Definition
For a function ρ : R ∞ → R the weak sub-gradient of ρ in X is defined as follows: ▽ ρ ( X ) := { a ∈ A | ρ ( X + Y ) ≥ ρ ( X ) + h Y, a i , ∀ Y ∈ R ∞ } . (5.1)This set can be empty.We have the following theorem: Theorem 5.1
Let ρ be a coherent risk measure on R ∞ with the Fatouproperty, given by the family P ⊆ D σ . Then a ∈ ▽ ρ ( X ) iff − a ∈ P and ρ ( X ) = h X, g i . roof Repeat exactly the proof of Theorem 17, Section 8.2 [10]. (cid:3)
Remark 5.1
We should remind that we have the same given definitionand theorem for coherent risks on L ∞ .Now we give the following main result of this section which couldbe interpreted as James’s Theorem in our framework: Theorem 5.2
Let A be a convex, σ ( A , R ∞ ) -closed subset of D σ . A is compact for σ ( A , R ∞ ) iff for each member X ∈ R ∞ it gets itssupremum on A . Proof ( ⇒ ) is obvious.( ⇐ ). For the other direction define: ρ ( X ) := sup A h X, a i . (5.2)By Theorem 2.1 it is clear that ρ ( − X ) is a coherent risk with Fatouproperty. It is not difficult to see that Var( A ) is convex and weakclosed subset in L . Let X ∈ L ∞ be a constant random process. ByRemark 3.2 and relation 3.3, ¯ ρ ( X ) = sup f ∈ Var( A ) E [ Xf ]. By the assump-tion of the theorem there exists a ∈ A such that ρ ( X ) = h X, a i andconsequently ¯ ρ ( X ) = E [Var( a ) X ]. This fact with James’s Theoremimply that Var( A ) is weakly compact. Now by Lemma 4.2 we deduce A is compact for topology σ ( A , R ∞ ). (cid:3) As a direct consequence of Theorems 4.1,5.1 and 5.2 we have :
Theorem 5.3
Let ρ : R ∞ → R be a coherent risk measure given by P ⊆ A . Then for every X ∈ L ∞ , ▽ ρ ( X ) = ∅ iff P is σ ( A , R ∞ ) -compact (or Var ( P ) is σ ( L , L ∞ ) -compact) iff ρ (or ¯ ρ ) has Lebesgueproperty. And finally by a simple calculation and using Theorems 5.1,5.2 and5.3 we have:
Theorem 5.4 if X = X + ... + X N and if − Q ∈ ▽ ρ then the allo-cation k i = E Q [ − X i ] is a fair allocation . Conclusion.
We have defined the Lebesgue property for convex risk on boundedc`adl`ag processes and the static risk due to each convex risk. Wehave characterized Lebesgue property and we have shown that ρ hasLebesgue property iff the static risk measure ¯ ρ corresponded to thathas Lebesgue property. This extends the original definition and char-acterization for the Lebesgue property of [16]. Finally we solved theso-called allocation problem in the fair way in the sense of fuzzy gametheory. It is our belief that studying risk measures on R ∞ opens thedoor to interesting lines of research. The coherent risk measures on R ∞ with the Lebesgue property could be explored in connection withfuzzy game theory problems like those studied in [9]. Since the proof of theorem 4.2 is a rather technical and long we givethe proof in this part.
Proof of Theorem 4.2. (2) ⇔ (3). First of all we mention that a sub-set of L p is r.c. iff its absolute value is r.c. Actually for p = 1 thiscomes from the fact that bounded sets are r.c. sets and for p = 1, byDunford-Pettis theorem, uniformly integrable sets are r.c. sets. Let A ± = { a ± | a ∈ A } where a + , a − are increasing decomposition of a . Itis obvious that C ± = { ( a T − a ) ± | a ∈ A } = Var( A ± ). So we haveVar( A ) ⊆ | C | and C ⊆ | Var( A ) | . Now by above arguments theproof is complete.(1) ⇔ (2). We split this part into two cases. Case 1: p = 1.Consider F t = F . In this case by Theorems 65,67 of Section VII [11]we know that A p is the dual of R q . Since A p is endowed with theweak* topology then A is r.c. iff it is bounded and this is true iffVar( A ) is bounded or, in the other words, r.c. for topology σ ( L p , L q ).When F t is nontrivial A is a relatively compact set of ˆ A p . The asser-tion is true because of the continuity of Π ∗ . Case 2: p = 1.( ⇒ ): We claim that C ± are relatively compact. Let a λT − a λ be a netin C and X be a member of L ∞ . Then by the relative compactness f A there is a subnet a β and a such that a β σ ( A , R ∞ ) −−−−−−→ a . This gives: E [( a βT − a β ) X ] = h Π op ( X ) , a β i → h Π op ( X ) , a i = E [( a T − a ) X ] . That means C is r.c. for σ ( L , L ∞ ). By Dunford-Pettis theorem weknow that this is equivalent to saying that C is uniformly integrable.Then | C | = {| f || f ∈ C } is uniformly integrable and consequently C ± are uniformly integrable. Again by Dunford-Pettis C ± are r.c.Now by Var( A ) ⊆ Var( A + ) + Var( A − ) = C + + C − we get that Var( A )is r.c.( ⇐ ):We define a topology on R ∞ . For that we define the semi normswhich generate this topology.For any weakly relatively compact subset H in L let V ( H ) := { a ∈A |∃ f ∈ H s.t.Var( a ) ≤ | f |} . Now define the following semi norm for H on R ∞ : P H ( X ) = sup a ∈ V ( H ) h X, a i . This topology is compatible with the vector structure because obvi-ously the V ( H )’s are bounded. We show this topology by σ . Let( R ∞ ) ′ be the dual of R ∞ with respect to topology σ . It is clear that A ⊆ ( R ∞ ) ′ . We want to show that A = ( R ∞ ) ′ .Let µ be an arbitrarily element of ( R ∞ ) ′ and X n be a non-negativesequence such that ( X n ) ∗ P −→
0. Then by Theorem 2.4 we have :0 ≤ P H ( X n ) ≤ sup f ∈ H E [( X n ) ∗ | f | ] → . (7.1)This gives X n σ −→ µ ( X n ) →
0. This fact,and (5.1) ofChapter VII [11] show that any µ can be decomposed into a differenceof two positive functionals. Let µ + be the positive part. By definitionfor any X ≥ µ + ( X ) = sup ≤ Y ≤ X µ ( Y ). Let X n be a positive anddecreasing sequence for which ( X n ) ∗ ↓ ≤ Y n ≤ X n be such that µ + ( X n ) ≤ µ ( Y n ) + n . Then since ( Y n ) ∗ P −→ ≤ µ + ( X n ) ≤ µ ( Y n ) + 1 n → . By this fact and Theorem 2 of Chapter VII [11] we get that µ + ∈ A .Similarly µ − ∈ A so then µ ∈ A .That means A = ( R ∞ ) ′ .The Corollary to Mackey’s Theorem 9, Section 13, Chapter 2 [14] eads us to σ ⊆ τ ( R ∞ , A ), where τ ( R ∞ , A ) is the Mackey’s topol-ogy. By this relation we get that for a relatively weakly compactsubset H of L there exists C , a compact disk in ( A , σ ( A , R ∞ )), forwhich { X | sup a ∈ C h X, a i < } ⊂ { X | P H ( X ) ≤ } . By polarity V ( H ) ⊆{ X | P H ( X ) ≤ } ◦ ⊆ { X | sup a ∈ C h X, a i < } ◦ . Using the generalizedBourbaki-Alaoglu Theorem we get that { X | sup a ∈ C h X, a i < } ◦ is com-pact in the topology σ ( A , R ∞ ).Let H = Var( A ). By A ⊆ V (Var( A )), the proof is complete. (cid:3) References [1] Aliprantis ,C.D ; Burkinshaw ,o (1978) Locally Solid RieszSpaces . Academic Press , New York.[2] Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc;Heath, David
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