Topological duals of Banach function spaces
aa r X i v : . [ m a t h . P R ] A ug Topological duals of Banach function spaces
Teemu Pennanen ∗ Ari-Pekka Perkki¨o † August 21, 2020
Abstract
This paper studies topological duals of Banach function spaces (BFS).We assume a finite measure but our arguments extend to general locallyconvex function spaces whose topology is generated by seminorms thatsatisfy the usual BFS axioms. The dual is identified with the direct sumof another space of random variables (K¨othe dual), a space of purelyfinitely additive measures and the annihilator of L ∞ . In the special caseof rearrangement invariant spaces, the second component in the dual van-ishes and we obtain various classical as well as new duality results e.g.on Lebesgue, Orlicz, Lorentz-Orlicz spaces and spaces of finite moments.Beyond rearrangement invariant spaces, we find the topological duals ofMusielak-Orlicz spaces and those associated with general convex risk mea-sures. Keywords.
Banach function spaces, topological duals, finitely additive mea-sures
AMS subject classification codes.
Banach function spaces (BFS) provide a convenient set up for functional analysisin spaces of measurable functions. Many well known properties of e.g. Lebesguespaces and Orlicz spaces extend to BFS with minor modifications; see e.g. [13,23, 1, 9]. Much of the theory focuses on rearrangement invariant (ri) spaceswhere the value of the norm a function only depends on the distribution of thefunction. Such spaces are arguably the most important among BFS but theydo exclude some interesting cases such as Musielak-Orlicz spaces and spaces ofrandom variables that arise in the theory of risk measures; see Section 5 below.This paper studies the topological duals of locally convex spaces of randomvariables where the topology is generated by an arbitrary collection of semi-norms that satisfy the usual properties of BFS-norms; see Section 4 below. ∗ Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UnitedKingdom, [email protected] † Mathematics Institute, Ludwig-Maximilian University of Munich, Theresienstr. 39, 80333Munich, Germany, [email protected]. Corresponding author L ∞ , we identify the topological dual as the direct sum of another space ofrandom variables (K¨othe dual), a space of purely finitely additive measures andthe annihilator of L ∞ . The last two components have a singularity propertythat has been found useful in the analysis of convex integral functionals byRockafellar [19] in the case of L ∞ and by Kozek [8] in the case of Orlicz spaces.In the case of L ∞ , the last component in the dual vanishes while in other Or-licz spaces, the second one vanishes; see [17]. Our result thus unifies the twoseemingly complementary cases.The main result is illustrated by simple derivations of various existing dualityresults in Musielak-Orlicz, Marcinkiewich, Lorentz and Orlicz-Lorentz spaces.In the last case, we also obtain an expression for the dual norm which seemsto be new. We go beyond the existing BFS settings by identifying topologicalduals of the space of random variables with finite moments, generalized Orliczspaces as well as spaces of random variables associated with general convexrisk measures. Such spaces have attracted attention in the recent literature oninsurance and financial mathematics; see e.g. [16], [10] and [7].The last section establishes the necessity of our axioms for locally convexspaces of random variables that are in separating duality with an other one.More precisely, a complete solid decomposable space in separating duality withanother solid decomposable space has a compatible topology generated by acollection of seminorms satisfying the usual BFS-properties.The rest of the paper is organized as follows. Section 2 reviews the dual-ity theory for L ∞ . Section 3 extends the notion of an integral with respectto a finitely additive measure to measurable not necessarily bounded randomvariables. Section 4 defines a general locally convex space of random variablesand gives the main result of the paper by characterizing the topological dualof a space. Section 5 applies the main result to characterize the topologicaldual in various known and new settings. Section 6 concludes by illustrating thenecessity of the employed axioms. L ∞ Let (Ω , F , P ) be a probability space with a σ -algebra F and a countably additiveprobability measure P . This section gives a quick review of the Banach space L ∞ of equivalence classes of essentially bounded measurable functions on aprobability space (Ω , F , P ). We consider R n -valued functions and endow L ∞ with the norm k u k L ∞ := | ( k u k L ∞ , . . . , k u n k L ∞ ) | , where | · | is a norm on R n . The dual norm on R n is denoted by | · | ∗ .Let M be the set of P -absolutely continuous finitely additive R n -valuedmeasures on (Ω , F ) and let M s be set of those m s ∈ M which are singular (“purely finitely additive” in the terminology of [22]; see [22, Theorem 1.22]) inthe sense that there is a decreasing sequence ( A ν ) ∞ ν =1 ⊂ F with P ( A ν ) ց m s | ∗ (Ω \ A ν ) = 0. Given m ∈ M , the set function | m | ∗ : F → R is defined by | m | ∗ ( A ) := | m + ( A ) + m − ( A ) | ∗ , where i th components of m + ∈ M and m − ∈ M are the positive and negativeparts, respectively, of the i th component m i of m ; see [22, Theorem 1.12].Recall that the space E of R n -valued simple random variables (i.e. piecewiseconstant with a finite range) is dense in L ∞ . Given m ∈ M , the integral of a u ∈ E is defined by Z Ω udm := J X j =1 α j m ( A j ) , where A j ∈ F and α j ∈ R n , j = 1 , . . . , m are such that u = P mj =1 α j A j On L ∞ , the integral is defined as the unique norm continuous linear extension from E to L ∞ .The following is from [22, Theorem 2.3] except that we do not assume thatthe underlying measure space is complete. The proof uses [5, Theorem 20.35]which does not rely on the completeness but identifies the dual of L ∞ with thespace of finitely additive measures that are absolutely continuous with respectto P . Combined with the results of [22, Section 1] on decomposition of finitelyadditive measures then completes the proof. The extension to spaces of R n -valued random variables is straightforward. Theorem 1 (Yosida–Hewitt) . The topological dual ( L ∞ ) ∗ of L ∞ can be identi-fied with M in the sense that for every u ∗ ∈ ( L ∞ ) ∗ there exist unique m ∈ M such that h u, u ∗ i = Z Ω udm, where the integral is defined componentwise. The dual norm is given by k m k ∗ L ∞ = | m | ∗ (Ω) . Moreover, M = L ⊕ M s in the sense that for every m ∈ M there existunique y ∈ L and m s ∈ M s such that Z Ω udm = E [ u · y ] + Z Ω udm s . We have m s = 0 if and only if h u A ν , u ∗ i → for every u ∈ L ∞ and everydecreasing ( A ν ) ∞ ν =1 ⊂ F such that P ( A ν ) ց .Proof. Assume first that n = 1. By [5, Theorem 20.35], the dual of L ∞ canbe identified with the linear space of finitely additive P -absolutely continuousmeasures m in the sense that every u ∗ ∈ ( L ∞ ) ∗ can be expressed as h u, u ∗ i = Z Ω udm L ∞ ) ∗ . By [22, Theorem 1.24],there is a unique decomposition m = m a + m s , where m a is countably additiveand m s is purely finitely additive. The construction in [22] also shows that m a and m s are absolutely continuous with respect to m and thus, absolutelycontinuous with respect to P as well. By [22, Theorem 1.22], there is a decreas-ing sequence ( A ν ) ∞ ν =1 ⊂ F such that P ( A ν ) ց m s (Ω \ A ν ) = 0. Thefunctional y s ∈ ( L ∞ ) ∗ given by h u, y s i := Z Ω udm s then has the property in the statement. By Radon-Nikodym, there exists a y ∈ L such that h u, u ∗ i := E [ u · y ] + Z Ω udm s . To prove the last claim, let u ∗ ∈ ( L ∞ ) ∗ and consider the representation interms of y ∈ L and m s ∈ M s given by the second claim. Let A ν be the sets inthe characterization of the singularity of m s . By [22, Theorems 1.12 and 1.17], m s = m s + − m s − for nonnegative purely finitely additive m s + and m s − . Given ǫ >
0, [22, Theorem 1.21] gives the existence of A ∈ F such that m s + (Ω \ A ) < ǫ and m s − ( A ) < ǫ . We have h u A A ν , u ∗ i = E [1 A A ν u · y ] + m s ( A ∩ A ν ) → m s ( A ) > m s + (Ω) − ǫ. By assumption, the left side converges to zero. Since ǫ > m s + = 0. By symmetry, we must have m s − = 0 so that m s = 0 which meansthat u ∗ is τ ( L ∞ , L )-continuous.By [22, Theorem 2.3], the dual norm of k · k L ∞ is given by k m k T V := m + (Ω) + m − (Ω). This completes the proof of the case n = 1. The general casefollows from the fact that the dual of a Cartesian product of Banach spaces isthe Cartesian product of the dual spaces with the norm k u k ∗ L ∞ = | ( k m k T V , . . . , k m n k T V ) | ∗ , which completes the proof. In [22] and in Section 2, integrals with respect to an m ∈ M were definedonly for elements of L ∞ as norm-continuous extensions of integrals of simplefunctions. Weakening the topology, it is possible to extend the definition ofthe integral to a larger space of measurable functions using Daniell’s construc-tion much as in [2, Chapter II] who considered countably additive integrals ofarbitrary (not necessarily F -measurable) functions.Another approach to integration of unbounded functions with respect tofinitely additive measures is that of Dunford; see Dunford and Schwartz [3] or4uxemburg [12]. A benefit of the Daniell extension adopted here is that it givesrise to a simpler definition of integrability that is easier to verify for larger classesof measurable functions.Given m ∈ M , we define ρ m : L → R by ρ m ( u ) := sup u ′ ∈ L ∞ (cid:26)Z Ω u ′ dm (cid:12)(cid:12)(cid:12)(cid:12) | u ′ j | ≤ | u j | ∀ j = 1 , . . . , n (cid:27) . We denote dom ρ m := { u ∈ L | ρ m ( u ) < ∞} . Lemma 2.
The function ρ m is a seminorm on dom ρ m and | Z Ω udm | ≤ ρ m ( u ) for all u ∈ L ∞ . For every u ∈ dom ρ m and ǫ > , there exists a u ′ ∈ L ∞ suchthat ρ m ( u − u ′ ) ≤ ǫ .Proof. We have ρ m ( u ) := n X j =1 ρ m j ( u j )where ρ m j ( u j ) = sup u ′ ∈ L ∞ ( R ) { Z Ω u ′ dm j | | u ′ | ≤ | u j |} . Thus we may assume that n = 1 and the claims from Theorem 27 in the ap-pendix.By Lemma 2, the integral is ρ m -continuous on L ∞ and L ∞ is ρ m -densein dom ρ m . Thus the integral has a unique ρ m -continuous linear extension todom ρ m . We call the extension the m -integral of u and denote it by Z Ω udm. The elements of dom ρ m will be said to be m -integrable . If m is countablyadditive, then, e.g., by the interchange rule [21, Theorem 14.60], ρ m ( u ) = n X j =1 Z Ω | u j | d | m j | = n X j =1 E [ | u j || y j | ] , where y is the density of m , and thus,dom ρ m = { u | u j ∈ L (Ω , F , | m j | ) ∀ j = 1 , . . . , n } . In this case, the integral is the Lebesgue integral.5
Topological duals of spaces of random vari-ables
This section presents the main results of the paper. The set up extends thatof Banach function spaces by replacing the norm by an arbitrary collection ofseminorms thus covering more general locally convex spaces of random variables.The main result identifies the topological dual of the space with the direct sumof the K¨othe dual and two spaces of singular functionals, the first of whichis represented by finitely additive measures while the second is the orthogonalcomplement of L ∞ .Let P be a collection of sublinear symmetric functions p : L → R , define U := \ p ∈P dom p, and endow U with the locally convex topology generated by P . Our aim is tocharacterize the topological dual U ∗ of U . To this end, we will assume that(A1) the topology of U is no weaker than the relative L -topology,and that each p ∈ P satisfies(A2) there exists a constant c such that p ( u ) ≤ c k u k L ∞ for all u ∈ L ,(A3) p ( u ′ ) ≤ p ( u ) for every u ∈ U and u ′ ∈ L with | u ′ | ≤ | u | .Occasionally, we will also assume the following(A4) p ( u A ν ) ց u ∈ L ∞ and decreasing sequence ( A ν ) ∞ ν =1 ⊂ F with P ( A ν ) ց p ( u A ν ) ց u ∈ U and decreasing sequence ( A ν ) ∞ ν =1 ⊂ F with P ( A ν ) ց p ( u ν ) ց u ν ) ∈ L ∞ such that | u ν | ց p ( u ν ) ց u ν ) ∈ U such that | u ν | ց P is a singleton, U is a normed space and we are in the setting ofnormed K¨othe function spaces; see e.g. [23]. If, in addition, p is lower semicon-tinuous in L , then U is a Banach function space; see Remark 4 below. In theBanach space setting, (A1) and (A2) hold under (A5’) if U has a weak unit;see e.g. [11, Theorem 1.b.14]. Necessity of the axioms will be discussed in moredetail in Section 6. 6 emark 3. Given p ∈ P , we define, ˆ φ p ( t ) := sup A ∈F { p (1 A ) | P ( A ) ≤ t } , ˇ φ p ( t ) := inf A ∈F { p (1 A ) | P ( A ) ≥ t } . Since we assume (A2) and (A3), the condition (A4) is equivalent to lim t ց ˆ φ p ( t ) = 0 . If lim t ց ˇ φ p ( t ) > , then U = L ∞ .Assume now that p is rearrangement invariant in the sense that p ( u ) = p (˜ u ) whenever u and ˜ u have the same distribution. Then, for any A ∈ F with P ( A ) = t , ˆ φ p ( t ) = ˇ φ p ( t ) = p (1 A ) where the common value is known as the fundamental function . In particular, dom p = L ∞ if (A4) does not hold while (A4) is equivalent to lim t ց ˆ φ p ( t ) = 0 .Proof. Assuming (A4), let t ν ց
0. There exists ( A ν ) ∞ ν =1 such that P ( A ν ) ≤ t ν and ˆ φ p ( t ν ) ≤ p (1 A ν ) + 1 /ν . Passing to a subsequence if necessary, 1 A ν → A ν := S ν ′ ≥ ν A ν , ( ˆ A ν ) ∞ ν =1 is decreasing with A ν ⊂ ˆ A ν and P ( ˆ A ν ) ց
0, so, by (A3)–(A4)ˆ φ ( t ν ) ≤ p (1 ˆ A ν ) + 1 /ν ց . For the converse, let u ∈ L ∞ and ( A ν ) ∞ ν =1 ⊂ F with t ν := P ( A ν ) ց
0. By(A3), p ( u νA ) ≤ k u k L ∞ ˆ φ ( t ν ) ց t ց ˇ φ p ( t ) > δ for some δ >
0, then p ( u ) ≥ p ( ν | u |≥ ν ) ≥ δν whenever P ( {| u | ≥ ν } ) >
0, so p ( u ) = + ∞ if u / ∈ L ∞ . Remark 4.
As soon as (A1) holds, (relative) weak compactness and sequential(relative) weak compactness on U are equivalent (Eberlein–Smulian property).If, in addition, p are lower semicontinuous on L , then U is complete. In thiscase, U is a Banach/Fr´echet space if P is a singleton/countable.Denoting p ( u ) = ρ ( | u | ) , the function p is lsc in L if and only if ρ has theFatou property: for any sequence ( η ν ) ∞ ν =1 ⊂ L with η ր η ∈ L , lim ρ ( η ν ) = ρ ( η ) .Proof. The first claim follows from the Theorem on p. 31 and Remark (2) onp. 39 in [4]. If ( u ν ) is a Cauchy net in U , it is Cauchy also in L so it L -converges to an u ∈ L . Being Cauchy in U means that for every ǫ > p ∈ P , there is a ¯ ν such that p ( u ν − u µ ) ≤ ǫ ∀ ν, µ ≥ ¯ ν. The lower semicontinuity then gives p ( u ν − u ) ≤ ǫ ∀ ν ≥ ¯ ν u ∈ U , by triangle inequality, and ( u ν ) converges in U to u . Thus U iscomplete.If p is lsc, lim inf ρ ( η ν ) ≥ ρ ( η ) while (A3) gives lim sup ρ ( η ν ) ≤ ρ ( η ). IfFatou property holds and u ν → u in L , then, passing to a subsequence ifnecessary, ¯ η ν := inf ν ′ ≥ ν | u ν ′ | increases pointwise to | u | , so p ( u ) = lim inf ρ (¯ η ν ) ≤ lim inf p ( u ν ). Remark 5.
Under (A2) and (A3), U is solid and decomposable. Solidity meansthat u ∈ U , u ′ ∈ L and | u ′ | ≤ | u | imply u ′ ∈ U . Decomposability means that u A + ¯ u Ω \ A ∈ U for every u ∈ U , ¯ u ∈ L ∞ and A ∈ F .Proof. Assumption (A2) implies that L ∞ ⊂ U while (A3) gives u A ⊂ U when-ever A ∈ F and u ∈ U . Since U is a linear space, the claim follows.For each p ∈ P , we define a sublinear symmetric function p ◦ on M by p ◦ ( m ) := sup u ∈ L ∞ (cid:26)Z Ω udm | p ( u ) ≤ (cid:27) . Lemma 6.
Let p ∈ P . For each m ∈ dom p ◦ , every u ∈ dom p is m -integrableand Z Ω udm ≤ p ( u ) p ◦ ( m ) . For every m ∈ dom p ◦ , there exist unique y ∈ L ∩ dom p ◦ and m s ∈ M s ∩ dom p ◦ such that Z Ω udm = E [ u · y ] + Z Ω udm s ∀ u ∈ dom p. Given m s ∈ M s ∩ dom p ◦ , there exists a decreasing ( A ν ) ∞ ν =1 ⊂ F such that P ( A ν ) ց and Z u Ω \ A ν dm s = 0 for every u ∈ dom p . Under (A4), M s ∩ dom p ◦ = { } .Proof. Lemma 2 and (A3) give Z Ω udm ≤ ρ m ( u ) ≤ sup u ′ ∈ L ∞ { Z Ω u ′ dm | | u ′ | ≤ | u |} ≤ p ( u ) p ◦ ( m ) . By Theorem 1, there exist y ∈ L and m s ∈ ( L ∞ ) s such that m = yP + m s .Let α < p ◦ ( y ) and α s < p ◦ ( m s ) and u, u s ∈ L ∞ such that p ( u ) , p ( u s ) ≤ Z Ω uydP ≥ α and Z Ω u s dm s ≥ α s . Let ( A ν ) ∞ ν =1 ⊂ F be decreasing with P ( A ν ) ց m s (Ω \ A ν ) = 0 and let u ν = λu Ω \ A ν + (1 − λ ) u s A ν , where λ ∈ (0 , p ( u ν ) ≤ λp ( u Ω \ A ν ) + (1 − λ ) p ( u s A ν ) ≤ λp ( u ) + (1 − λ ) p ( u s ) ≤ Z Ω u ν dm ≥ λα + (1 − λ ) α s . Thus, p ◦ ( m ) ≥ λα +(1 − λ ) α s . Since α < p ◦ ( y ) and α s < p ◦ ( m s ) were arbitrary, p ◦ ( m ) ≥ λp ◦ ( y ) + (1 − λ ) p ◦ ( m s ). Since λ ∈ (0 ,
1) was arbitrary, we get p ◦ ( y ) ≤ p ◦ ( m ) and p ◦ ( m s ) ≤ p ◦ ( m ). Thus, y ∈ dom p ◦ and m s ∈ dom p ◦ .To prove the last claim, let m s ∈ M s ∩ dom p ◦ . By the first claim, Z Ω u A dm s ≤ p ( u A ) p ◦ ( m s ) ∀ u ∈ L ∞ , A ∈ F so, by the last claim of Theorem 1, condition (A4) implies m s = 0.Let M be the set of P -absolutely continuous finitely additive measures m such that p ◦ ( m ) < ∞ for some p ∈ P . The set of purely finitely additiveelements of M will be denoted by M s . The set of densities y = dm/dP ofcountably additive m ∈ M will be denoted by Y .The following is the main result of this section. It identifies the topologi-cal dual of U with the direct sum of the K¨othe space, purely finitely additivemeasures M s and the annihilator( L ∞ ) ⊥ := { w ∈ U ∗ | h u, w i = 0 ∀ u ∈ L ∞ } of L ∞ . Theorem 7.
We have U ∗ = Y ⊕ M s ⊕ ( L ∞ ) ⊥ in the sense that for every u ∗ ∈ U ∗ there exist unique y ∈ Y , m s ∈ M s and w ∈ ( L ∞ ) ⊥ such that h u, u ∗ i = E [ u · y ] + Z Ω udm s + h u, w i . For every u ∈ U and m ∈ M , Z Ω udm ≤ p ( u ) p ◦ ( m ) . Given w ∈ ( L ∞ ) ⊥ and u ∈ U , there exists a decreasing sequence ( A ν ) ∞ ν =1 ⊂ F with P ( A ν ) ց and h u, w i = h u A ν , w i ∀ ν = 1 , , . . . . Under (A4), M s = { } and under (A5), ( L ∞ ) ⊥ = { } .Proof. By Lemma 6,
M ⊂ U ∗ , so M ⊕ ( L ∞ ) ⊥ ⊆ U ∗ . To prove the oppositeinclusion, let u ∗ ∈ U ∗ . There exists p ∈ P and γ > u ∗ ≤ γp .Assumption (A2) implies that u ∗ is continuous in L ∞ . By Theorem 1, thereexists a unique m ∈ M such that h u, u ∗ i = R Ω udm for all u ∈ L ∞ . Since9 ∗ ≤ γp , we have m ∈ dom p ◦ , so m is continuous on U by Lemma 6. Now w := u ∗ − m belongs to ( L ∞ ) ⊥ , so u ∗ has the required decomposition. Givenanother decomposition u ∗ = ˜ m + ˜ w with ˜ w ∈ ( L ∞ ) ⊥ and ˜ m ∈ M , we have( m − ˜ m ) + ( w − ˜ w ) = 0. Thus R Ω ud ( m − ˜ m ) = 0 for all u ∈ L ∞ , so m − ˜ m = 0and hence also w − ˜ w = 0, so the decomposition is unique.The inequality follows directly from that of Lemma 6. Let u ∈ U and A ν := {| u | > ν } . Clearly P ( A ν ) ց u Ω \ A ν ∈ L ∞ , so h u Ω \ A ν , w i = 0and thus w is singular. That M = Y under (A4) is the last claim of Lemma 6.Under (A5), the truncations u ν := u {| u |≥ ν } of any u ∈ U converge to u so L ∞ is dense in U and thus, ( L ∞ ) ⊥ = { } .Applications of Theorem 7 are given in Section 5. When P is a singleton,we are in the setting of [23], where the dual of U is decomposed into a directsum of Y and ”singular elements”. Theorem 7 gives a more precise descriptionof the singular elements as a direct sum of M s and ( L ∞ ) ⊥ .Note that the inequality in Theorem 7 implies that p ◦ coincides on M withthe polar (i.e., the dual seminorm) of p . An application of Theorem 7 and theHahn-Banach theorem gives the following result, where ˜ U is the closure of L ∞ in U . Corollary 8.
We have ˜ U ∗ = M in the sense that for every ˜ u ∗ ∈ ˜ U ∗ there exist unique m ∈ M such that h ˜ u, ˜ u ∗ i = Z Ω ˜ udm. In particular, if (A4) holds, then ˜ U ∗ = Y and if (A5) holds, then ˜ U = U . The following lists some basic properties of the K¨othe dual Y . Lemma 9.
We have1. L ∞ ⊆ Y and, for each p ∈ P
2. there is a constant c such that c k y k L ≤ p ◦ ( y ) for all y ∈ L ,3. p ◦ ( y ′ ) ≤ p ◦ ( y ) for every y ′ , y ∈ L with | y ′ | ≤ | y | .4. We have the “H¨older’s inequality” E [ u · y ] ≤ p ( u ) p ◦ ( y ) and, conversely, if there is c > such that c k u k L ≤ p ( u ) for all u and p is lsc in L , then p ◦ ( y ) < ∞ whenever E [ u · y ] < ∞ for all u ∈ dom p .In particular, Y is solid and decomposable. roof. Assumption (A1) implies 1, and (A2) implies 2. By (A3), p ◦ ( y ′ ) = sup u ′ ∈ L ∞ ,u ∈ L ∞ { E [ u ′ · y ′ ] | | u ′ | ≤ | u | , p ( u ) ≤ } = sup u ∈ L ∞ { E [ | u || y ′ | ] | p ( u ) ≤ }≤ sup u ∈ L ∞ { E [ | u || y | ] | p ( u ) ≤ } = p ◦ ( y ) , so 3 holds. To prove 4, the inequality in Lemma 6 gives the H¨older’s inequality.Assume now that p ◦ ( y ) = + ∞ . Let α ν > P α ν = 1. There exists u ν with p ( u ν ) ≤ u ν · y ≥ E [ u ν · y ] ≥ /α ν . We have that P νν ′ =1 α ν ′ u ν ′ converges to u := P α ν u ν in L and, since p is lsc in L , u ∈ dom p . Bymonotone convergence, E [ u · y ] = ∞ X ν =1 α ν E [ u ν · y ] = + ∞ , which completes the proof. The following example is a direct application of Corollary 8.
Example 10 (The space of finite moments) . The L p -norms with p = 1 , , . . . satisfy (A1)-(A5), so the space U := \ p ≥ L p of measurable functions with finite moments is a Fr´echet space and its dual maybe identified with Y := [ p> L p under the bilinear form h u, y i = E [ u · y ] . –Spaces with finite moments strictly less that p ??Given a set C in a linear space, we will use the notationpos C := [ α> ( αC ) and C ∞ := \ α> ( αC ) . The following construction, inspired by the Luxemburg norm in the theory ofOrlicz spaces, turns out to be convenient.
Example 11.
Let H : L → R + be lsc convex such that H (0) = 0 and H1) there is a constant c > such that H ( u ) ≤ implies k u k L ≤ c ,(H2) L ∞ ⊂ pos(dom H ) ,(H3) H ( u ) ≤ H ( u ) whenever | u | ≤ | u | .The function p ( u ) := inf { β > | H ( u/β ) ≤ } is lsc, symmetric and sublinear. Let P = { p } and U = dom p . Assumptions(A1)–(A3) hold and, in particular, U is a Banach space with dual U ∗ = M ⊕ ( L ∞ ) ⊥ , where M = pos dom H ∗ with H ∗ : M → R given by H ∗ ( m ) := sup u ∈ L ∞ { Z Ω udm − H ( u ) } . For any m ∈ M , p ◦ ( m ) = sup u ∈ L ∞ { Z Ω udm | H ( u ) ≤ } = inf β> { βH ∗ ( m/β ) + β } , restriction of p ◦ to M is the polar of p and k m k H ∗ ≤ p ◦ ( m ) ≤ k m k H ∗ , where k m k H ∗ := inf { β > | H ∗ ( m/β ) ≤ } . Assume now that L ∞ ⊆ dom H . If(H4) H ( u ν ) ց whenever ( u ν ) ∞ ν =1 ⊂ L ∞ with | u ν | ց almost surely,then (A4) holds so M s = { } and the dual of the closure ˜ U of L ∞ in U can beidentified with Y . If(H5) H ( u ν ) ց whenever ( u ν ) ∞ ν =1 ⊂ dom H with | u ν | ց almost surely,then ˜ U = (dom H ) ∞ . In particular, U = ˜ U if dom H is a cone.Proof. Let u ν → u in L be such that p ( u ν ) ≤ α or, in other words, H ( u ν /α ) ≤
1. Thus lower semicontinuity of H implies that of p . It is clear that (H1)implies (A1). By (H2), p is finite on L ∞ . Since p is lsc on L , it is lsc on σ ( L ∞ , L ). Thus, by [20, Corollary 8B], p is continuous in L ∞ and thus (A2)holds. Assumption (A3) is clear from (H3).12et m ∈ M . Since the infimum in the definition of the Luxemburg norm isattained, p ◦ ( m ) = sup u ∈ L ∞ { Z Ω udm | p ( u ) ≤ } = sup u ∈ L ∞ { Z Ω udm | H ( u ) ≤ } . Lagrangian duality gives p ◦ ( m ) = inf β> sup u ∈ L ∞ { Z Ω udm − βH ( u ) + β } = inf β> { βH ∗ ( m/β ) + β } . Clearly, p ◦ ( m ) ≤ inf β> { βH ∗ ( m/β ) + β | H ∗ ( m/β ) ≤ } ≤ { β > | H ∗ ( m/β ) ≤ } . On the other hand, we have p ◦ ( m ) = inf β> { βH ∗ ( m/β ) + β } = inf α> g ( αm ) α , where g ( m ) = H ∗ ( m ) + 1. Since H ∗ ≥
0, we have g ≥ k · k H ∗ when k m k H ∗ ≤ k m k H ∗ >
1, convexity and the fact that H ∗ (0) = 0 give H ∗ ( m/ k m k H ∗ ) ≤ H ∗ ( m ) / k m k H ∗ . By definition of k m k H ∗ , the left side equals 1 so k m k H ∗ ≤ H ∗ ( m ) ≤ g ( m ).Thus, p ◦ ( m ) ≥ inf α> k αm k H ∗ α = k m k H ∗ . If (H4) holds and | u ν | ց L ∞ , then for all β > H ( u ν /β ) ց p ( u ν ) ց
0. In particular, (A4) holds.To prove the last claim, let u ∈ (dom H ) ∞ , u ν := u | u |≤ ν and β >
0. By(H3), u − u ν = u Ω \{| u |≤ ν } ∈ β dom H so (H5) implies H (( u − u ν ) /β ) ց . Since β > p ( u − u ν ) ց H ) ∞ ⊆ ˜ U . To provethe converse, it remains to show that (dom H ) ∞ is closed in U . If ( u ν ) is in(dom H ) ∞ and converges to u ∈ ˜ U , we have for any β > H ( u/ (2 β )) ≤ H ( u ν /β ) + 12 H (( u − u ν ) /β ) ≤ H ( u ν /β ) + 12for ν large enough, so H ( u/ β ) < ∞ and thus u ∈ (dom H ) ∞ .13usielak–Orlicz spaces are generalizations of Orlicz spaces where the associ-ated Young function Φ is allowed to be random in the sense that it is a functionon R × Ω such that ω
7→ { ( ξ, α ) | Φ( ξ, ω ) ≤ α } is a convex-valued measurable mapping; see [21, Chapter 14]. If Φ only takesfinite real values, this happens exactly when Φ( ξ, · ) is measurable for every ξ ∈ R and Φ( · , ω ) is convex for every ω ∈ Ω. The dual of a Musielak–Orlicz space canbe characterized in terms of the conjugate function defined byΦ ∗ ( η, ω ) = sup ξ ∈ R { ξη − Φ( ξ, ω ) } . The measurability condition on Φ implies the same property for Φ ∗ ; see [21,Theorem 14.50]. Example 12 (Musielak-Orlicz spaces) . Let
Φ : R × Ω → R + be nonzero randomsymmetric convex function with Φ(0) = 0 and such that Φ( a, · ) , Φ ∗ ( a, · ) ∈ L for some constant a > . Endowed with the Luxemburg norm k u k Φ := inf { β > | E Φ( | u | /β ) ≤ } ,L Φ := { u ∈ L | k u k Φ < ∞} is a Banach space. The dual of L Φ is ( L Φ ) ∗ = L Φ ∗ ⊕ M s ⊕ ( L ∞ ) ⊥ , where M s = { m ∈ M s | σ Φ ( m ) < ∞} with σ Φ ( m ) := sup u ∈ L ∞ { R Ω udm | E Φ( | u | ) < ∞} . For any y + m s ∈ L Φ ∗ ⊕ M s ,the dual norm can be expressed as k y + m s k ∗ Φ = sup u ∈ L ∞ { E [ u · y ] + Z Ω udm s | E Φ( | u | ) ≤ } = inf β> { βE Φ ∗ ( | y | ∗ /β ) + β } + σ Φ ( m s ) , we have k y k Φ ∗ ≤ k y k ∗ Φ ≤ k y k Φ ∗ ∀ y ∈ L Φ ∗ , and the dual of the closure M Φ of L ∞ in L Φ is ( M Φ ) ∗ = L Φ ∗ ⊕ M s . Assume now that Φ( a, · ) ∈ L for all a > . Then, M s = { } , M Φ coincideswith the Morse heart (dom E Φ) ∞ = { ξ ∈ L | E Φ( | ξ | /β ) < ∞ ∀ β > } , and, in particular, L Φ = M Φ if dom E Φ is a cone. roof. We apply Example 11 to H ( u ) := E Φ( | u | ). By [21, Theorem 14.60], H ( u ) = sup η ∈ L ∞ E { [ | u | η ] − Φ ∗ ( η ) } , so H is L -lsc. This also gives H ( u ) ≥ a k u k L − E Φ ∗ ( a )so Φ ∗ ( a ) ∈ L implies (H1). The assumption Φ( a ) ∈ L implies that H ( u ) < ∞ when k u k L ∞ ≤ a so (H2) holds. Property (H3) holds since Φ is increasing. By[19, Theorem 1] and [18, Theorem 15.3], H ∗ ( m ) = sup u ∈ L ∞ { Z udm − Eh ( u ) } = E Φ ∗ ( | y | ∗ ) + σ Φ ( m s ) . If Φ( a ) ∈ L for all a >
0, then L ∞ ⊂ dom H and (H4) and (H5) hold bymonotone convergence theorem. Thus all the claims follow from Example 11.In [14], the assumption Φ( a, · ) ∈ L for all a > Example 13 (Risk measures) . Let ρ : L → R be a “convex risk measure” inthe sense that it is convex, nondecreasing, ρ (0) = 0 and ρ ( ξ + α ) = ρ ( ξ ) + α for all ξ ∈ L and α ∈ R . Assume that n = 1 , ρ is L -lsc and that there is aconstant c > such that ρ ( | u | ) ≤ implies k u k L ≤ c .Endowed with the norm k u k ρ := inf { β > | ρ ( | u | /β ) ≤ } ,L ρ := { u ∈ L | ρ ( | u | ) < ∞} is a Banach space whose dual can be identifiedwith M ⊕ ( L ∞ ) ⊥ , where M = { m ∈ M | ∃ β > α ( | m | /β ) < ∞} with α : M → R defined by α ( m ) := sup ξ ∈ L ∞ + { Z Ω ξdm − ρ ( ξ ) } . For any m ∈ M , the dual norm can be expressed as k m k ∗ ρ = sup u ∈ L ∞ { Z Ω udm | ρ ( u ) ≤ } = inf β> { βα ( | m | /β ) + β } , and k m k α ≤ k m k ∗ ρ ≤ k m k α , where k m k α := inf { β > | α ( | m | /β ) ≤ } . . If ρ has the Lebesgue property on L ∞ : ρ ( ξ ν ) ց for any decreasing se-quence ( ξ ν ) ⊂ L ∞ with ξ ν ց almost surely,then the dual of the closure ˜ L ρ of L ∞ in L ρ can be identified with L α := { y ∈ L | ∃ β > α ( | y | /β ) < ∞} .
2. If ρ has the Lebesgue property on dom ρ : ρ ( ξ ν ) ց for any decreasingsequence ( ξ ν ) ⊂ dom ρ with ξ ν ց almost surely,then ˜ L ρ = { u ∈ L | ρ ( | u | /β ) < ∞ ∀ β > } , and, in particular, L ρ = ˜ L ρ if dom ρ is a cone.Proof. We apply Example 11 to the function H ( u ) := ρ ( | u | ). By assumption,(H1) and (H3) hold. By monotonicity and translation invariance, ρ ( | u | ) ≤ ρ ( k u k L ∞ ) = k u k L ∞ , so L ∞ ⊂ dom H . In particular, (H2) holds. The conditions(H4) and (H5) in Example 11 translate directly to those of 1 and 2. Thus theclaims follow from Example 11, since here H ∗ ( m ) := sup u ∈ L ∞ { Z udm − ρ ( | u | ) } = sup u ∈ L ∞ ,ξ ∈ L ∞ + { Z uξdm − ρ ( ξ ) | | u | = 1 } = sup ξ ∈ L ∞ + { Z ξd | m | − ρ ( ξ ) } = α ( | m | ) , where the second last equality follows from [22, Theorem 2.3] and the fact that ν ( A ) := R A ξdm is a finitely additive measure with | ν | ( A ) = R A ξd | m | .Given u ∈ L , let n u ( τ ) := E {| u | >τ } and q u ( t ) := inf { τ ∈ R | n u ( τ ) ≤ t } . Note that τ − n u ( τ ) is the cumulative distribution function of | u | and that q u is an inverse of n u . Both n u and q u are nonincreasing. Lemma 14.
We have Z t q u ( t ) dt = inf s ∈ R + { ts + E [ | u | − s ] + } . Proof.
By Theorems 23.5 and 24.2 of [18], the functions f ( t ) := Z t q u ( s ) ds f ∗ ( s ) = Z s n u ( τ ) dτ − Z ∞ n u ( τ ) dτ = − Z ∞ s n u ( τ ) dτ are concave and conjugate to each other. By Fubini, f ∗ ( s ) = − E Z ∞ s {| u | >τ } dτ = − E [ | u | − s ] + so Z t q u ( s ) ds = inf s ∈ R + { ts + E [ | u | − s ] + } , by the biconjugate theorem (see e.g. [18, Theorem 12.2]).Recall that a probability space is resonant if it is atomless or completelyatomic with all atoms having equal measure. Example 15 (Lorentz and Marcinkiewicz spaces) . Assume that (Ω , F , P ) isresonant. Given a nonnegative concave increasing function φ on [0 , with φ (0) = 0 , the associated Marcinkiewicz space is the linear space M φ of u ∈ L with k u k φ := sup t ∈ (0 , (cid:26) φ ( t ) Z t q u ( s ) ds (cid:27) < ∞ . The function k · k φ is a norm and M φ is a Banach space. If lim t ց t/φ ( t ) > ,we have M φ = L ∞ . Assume now that lim t ց t/φ ( t ) = 0 . The topological dualof M φ is M ∗ φ = Λ Φ ⊕ ( L ∞ ) ⊥ , where Λ Φ is the Lorentz spaceΛ φ := { y ∈ L | k y k ∗ φ < ∞} , where k y k ∗ φ := Z q y ( t ) dφ ( t ) . The closure of L ∞ in M φ can be expressed as ˜ M φ = { u ∈ L | lim t ց φ ( t ) Z t q u ( s ) ds = 0 } . The topological dual of ˜ M φ is Λ Φ and the topological dual of Λ φ is M φ .Proof. By Lemma 14, u Z t q u ( t ) dt is the infimal projection of a sublinear function of s and u and thus, sublinearin u . It is also continuous in L . It follows that k · k φ is sublinear, symmetricand lsc in L . 17ince k u k φ ≥ φ (1) Z q u ( s ) ds = φ (1) E [ | u | ] , (A1) holds. By Remark 4, M φ is Banach. Since q u ≤ k u k L ∞ , we have k u k φ ≤ sup t ∈ (0 , tφ ( t ) k u k L ∞ , where sup t ∈ (0 , tφ ( t ) < ∞ since φ is concave and strictly positive for t > A ∈ F , k A k φ = sup t φ ( t ) min { t, P ( A ) } = P ( A ) φ ( P ( A )) , since t tφ ( t ) is increasing by concavity. Thus ˆ φ p ( t ) := tφ ( t ) is the fundamentalfunction of M φ . By Remark 3, M φ = L ∞ if lim t ց t/φ ( t ) > t ց t/φ ( t ) = 0. We have k y k ∗ φ = sup u ∈ L { E [ uy ] | k u k φ ≤ } = sup u ∈ L { Z q u ( t ) q y ( t ) dt | Z t q u ( s ) ds ≤ φ ( t ) ∀ t ∈ [0 , } = Z q y ( t ) φ ′ ( t ) dt = Z q y ( t ) dφ ( t ) , where the second equality follows from [1, Corollary 2.4.4] and the third fromHardy’s lemma [1, Proposition 2.3.6]. The representation of the topological dualof M φ now follows from Theorem 7.If u ∈ L ∞ , q u is bounded, solim t ց φ ( t ) Z t q u ( s ) ds = lim t ց tφ ( t ) 1 t Z [0 ,t ] q u ( s ) ds = 0 , by assumption. Thus, L ∞ ⊂ ˜ M φ . Let u ∈ M φ and ˜ M φ . We have q u +˜ u ( s + s ) ≤ q u ( s ) + q ˜ u ( s ), solim t ց φ ( t ) Z t q u ( s ) ds ≤ lim t ց φ ( t ) Z t ( q u − ˜ u ( s/
2) + q ˜ u ( s/ ds = lim t ց φ ( t ) Z t q u − ˜ u ( s/ ds = lim t ց φ ( t ) Z t q u − ˜ u ( s ) ds ≤ lim 1 φ (2 t ) Z t q u − ˜ u ( s ) ds ≤ k u − ˜ u k φ , φ . Thus, ˜ M φ is closedin M φ so ˜ M φ contains the closure of L ∞ . To prove the converse, let u ∈ ˜ M φ and u ν = u {| u |≤ ν } . We have q u − u ν ( t ) = 0 for t ≥ t ν := P ( | u | ≥ ν ) while q u − u ν ( t ) = q u ( t ) for t < t ν . Thus, k u − u ν k φ = sup t ∈ [0 , (cid:26) φ ( t ) Z t q u − u ν ( s ) ds (cid:27) = sup t ∈ [0 ,t ν ] (cid:26) φ ( t ) Z t q u ( s ) ds (cid:27) . Since u ∈ ˜ M φ , this converges to 0 as ν → ∞ . Thus, ˜ M φ is the closure of L ∞ in M φ .By Lemma 9, the Lorentz seminorm satisfies (A1)-(A3). If y ν ց k y ν k ∗ φ < ∞ , we have q y ν ց
0, so by monotone convergence, k y ν k ∗ φ ց
0. Thus,the Lorenz norm satisfies (A5). The fact that the topological dual of Λ φ is M φ now follows from Theorem 7 and the fact that, by the bipolar theorem, p is thepolar of p ◦ .–intersection of Markinkiewich spaces, Orlicz spaces?? Example 16 (Generalized Orlicz-spaces) . Let Φ be as in Example 12 with dom Φ = R and let r be a sublinear symmetric lsc function on L satisfying(A1)–(A4). Endowed with the norm k u k Φ ,r := inf { β > | r (Φ( | u | /β ) ≤ } , U := { u ∈ L | k u k Φ ,r < ∞} is a Banach space with dual U ∗ = Y ⊕ ( L ∞ ) ⊥ , where Y := { y ∈ L | k y k ∗ Φ ,r < ∞} with k y k ∗ Φ ,r = inf v ∈ L { E [ v Φ ∗ ( y/v )] + r ◦ ( v ) } . Moreover, k y k H ∗ ≤ k y k ∗ Φ ,r ≤ k y k H ∗ , where k y k H ∗ = inf { β > | H ∗ ( y/β ) ≤ } = inf v ∈ L max { r ◦ ( v ) , E [ v Φ ∗ ( y/v )] } . If r satisfies (A5), then the closure of L ∞ in U has the expression ˜ U = { u ∈ L | r (Φ( | u | /β )) < ∞ ∀ β > } . In this case, ˜ U = U if dom H is a cone. In particular, dom H is a cone if Φ satisfies ∆ -condition: there exists K > and x such that Φ(2 x ) ≤ K Φ( x ) forall x ≥ x . roof. This fits Example 11 with H ( u ) := ( r (Φ( | u | )) if Φ( | u | ) ∈ L , + ∞ otherwise . For every u ∈ L , H ( u ) = sup η ∈ L ∞ + { E [ η Φ( | u | )] − r ∗ ( η ) } , so H is lsc in L . Since r satisfies (A1)–(A4), H satisfies (H1)–(H4).We compute the conjugate of H by employing conjugate duality; see [20].Let F ( x, u ) := r (Φ( u )+ x ) be defined on L ∞ × L ∞ . The conjugate F ∗ on L × L has the expression F ∗ ( v, y ) := sup u,x ∈ L ∞ { E [ xv + uy ] − r (Φ( u ) + x ) } = sup u,x ∈ L ∞ { E [ vx − v Φ( u ) + uy ] − r ( x ) } = E [ v Φ ∗ ( y/v )] + δ B ∗ ( v ) , where the last equality comes from the interchange rule [21, Theorem 14.60] and B ∗ := { v ∈ L | r ◦ ( v ) ≤ } . Since r satisfies (A4), it is τ ( L ∞ , L )-continuous?. By [20, Theorem 17], thisimplies H ∗ ( y ) = inf v ∈ L F ∗ ( y, v ) = inf v ∈ L { E [ v Φ ∗ ( y/v )] | r ◦ ( v ) ≤ } so, by Example 11, k y k ∗ Φ ,r = inf β> { βH ∗ ( y/β ) + β } = inf β> ,v ∈ L { E [ βv Φ ∗ ( y/ ( βv ))] + β | r ◦ ( v ) ≤ } = inf β> ,v ∈ L { E [ v Φ ∗ ( y/v )] + β | r ◦ ( v ) ≤ β } = inf v ∈ L { E [ v Φ ∗ ( y/v )] + r ◦ ( v ) } . The claims concerning the dual space and its norm follow from Example 11. Wehave k y k H ∗ := inf { β > | H ∗ ( y/β ) ≤ } = inf { β > | ∃ v ∈ L : r ◦ ( v ) ≤ , E [ v Φ ∗ ( y/ ( βv ))] ≤ } = inf { β > | ∃ v ∈ L : r ◦ ( v ) ≤ β, E [ v Φ ∗ ( y/v )] ≤ β } = inf v ∈ L max { r ◦ ( v ) , E [ v Φ ∗ ( y/v )] } . r satisfies (A5). Then H satisfies (H5), so Example 11gives ˜ U = (dom H ) ∞ . The set on the right can be written as { u ∈ L | r (Φ( u/β ) < ∞ ∀ β > } .Note that if r is the L ∞ -norm, we simply have U = L ∞ and Y = L while if r is the L -norm, then we are back in Musielak-Orlicz spaces of Example 12. IfΦ is nonrandom and r is the Lorentz-norm associated with a concave function φ (see Example 15), U becomes the Orlicz-Lorentz-space studied e.g. in [6]. Inthis case the above expressions for the dual norm seem new. One could also take r the Marcinkiewicz norm in which case r ◦ is the Lorentz-norm. This settingseems new. This section goes beyond Banach and Fr´echet spaces. We assume that U and Y are solid decomposable spaces (see Remark 5) of random variables in separatingduality under the bilinear form h u, y i := E [ u · y ] . Clearly, solid spaces are decomposable but there are decomposable spacesthat are not solid.
Example 17.
Let (Ω , F ) := ([0 , , B ([0 , , u ( ω ) := ω − + ω − and U := L ∞ + Lin ( u A | A ∈ F ) . Then U is decomposable, by construction, but notsolid, since it does not contain ¯ u ( ω ) = ω − for which < ¯ u < u . The following two lemmas do not require solidity of U or Y . The first one isLemma 6 from [15]. Lemma 18.
We have L ∞ ⊆ U ⊆ L and σ ( L , L ∞ ) | U ⊆ σ ( U , Y ) , σ ( U , Y ) | L ∞ ⊆ σ ( L ∞ , L ) ,τ ( L , L ∞ ) | U ⊆ τ ( U , Y ) , τ ( U , Y ) | L ∞ ⊆ τ ( L ∞ , L ) . Lemma 19.
The following are equivalent:1. U is solid,2. y u · y is continuous from ( Y , σ ( Y , U )) to ( L , σ ( L , L ∞ )) ,3. η ηu is continuous from ( L ∞ , τ ( L ∞ , L )) to ( U , τ ( U , Y )) .Proof. For any u ∈ U , y ∈ Y and η ∈ L ∞ , E [( u · y ) η ] = E [( ηu ) · y ] . σ ( Y , U )-continuous in y if and only if there is a u ′ ∈ U such that E [( ηu ) · y ] = E [ u ′ · y ] for all y ∈ Y . Since L ∞ ⊂ Y separates the elements of L , weget that y E [( u · y ) η ] is continuous if and only if ηu ∈ U . This proves theequivalence of 1 and 2.Assume 2 and let K ⊂ Y be σ ( Y , U )-compact. We havesup y ∈ K h y, ηu i = sup y ∈ K h u · y, η i L ∞ = sup ξ ∈ D h ξ, η i L ∞ , where D = { u · y | y ∈ K } is σ ( L , L ∞ )-compact since y u · y is continuous. Corollary 20.
In the setting of Corollary 8, (A4) holds if and only if ˜ U ∗ = Y .Proof. By Lemma 6, (A4) implies M s = 0, so ˜ U ∗ = Y by Corollary 8. On theother hand, if ˜ U ∗ = Y , the topology of ˜ U cannot be stronger than τ ( ˜ U , Y ). Inthat case, Lemma 19 implies that p ( uη ν ) → η ν → τ ( L ∞ , L ). Since1 A ν → τ ( L ∞ , L ) if P ( A ν ) →
0, assumption (A4) holds.
Lemma 21.
A convex set C ⊂ U is σ ( U , Y ) -compact if and only if, for every y ∈ Y , the set { u · y | u ∈ C } is weakly compact in L .Proof. Since continuous images of compact sets are compact, Lemma 19 givesthe necessity. Let ( u ν ) be a net in C . Letting y range over unit constant vectors,we see that C is σ ( L , L ∞ )-compact. Thus there is a subnet and u ∈ C suchthat u ν → u in σ ( L , L ∞ ). Let y ∈ Y and ǫ >
0. Since { u · y | u ∈ C } isweakly compact in L , it is uniformly integrable, so there exists n such that | E [( u ν − u ) · y | y | >n ] | < ǫ for every ν . Since u ν → u in σ ( L , L ∞ ), there exists ν ′ such that | E [( u ν − u ) · y | y |≤ n ] | < ǫ for all ν ≥ ν ′ . Thus, for all ν ≥ ν ′ , | E [( u ν − u ) · y ] | ≤ ǫ, which proves that u ν → u in σ ( U , Y ) Corollary 22.
Given ¯ u ∈ U , the set C := { u ∈ U | | u | ≤ | ¯ u |} is σ ( U , Y ) -compact.Proof. By Lemma 21, it suffices to show that C y := { u · y | u ∈ U , | u | ≤ | ¯ u |} is σ ( L , L ∞ )-compact for every y ∈ Y . The set C y is uniformly integrable, so,by Dunford-Pettis, it suffices to show that C y is σ ( L , L ∞ )-closed. Since U issolid, C y = { u · y | u ∈ L , | u | ≤ | ¯ u |} . Let u ν · y → ξ in L , where | u ν | ≤ | ¯ u | . Passing to convex combinations, we mayassume, by Komlos lemma, that u ν → u almost surely for some u with | u | ≤ | ¯ u | .By dominated convergence, u ν · y → u · y in L , so C y is closed.22 heorem 23. If U is τ ( U , Y ) -complete, then there exists a collection P of lscsublinear symmetric functions p : L → R such that the topology generated by P on U is compatible with the duality, U = \ p ∈P dom p, Y = [ p ∈P dom p ◦ and each p ∈ P satisfies (A1)–(A5).Proof. Let C be the collection of σ ( Y , U )-compact solid convex subsets of Y andlet P the collection of the functions p : L → R of the form p ( u ) = sup y ∈ C E [ u · y ] , where C ∈ C . Each p ∈ P is convex and positively homogeneous. Since the unitball of L ∞ is in C , (A1) holds. The topology generated by P is weaker than theMackey-topology which is generated by all σ ( Y , U )-compact sets. By Lemma 18,(A2) holds. Given ¯ y ∈ Y , { y ∈ Y | | y | ≤ | ¯ y |} is compact by Corollary 22. It isalso solid and convex, so the topology generated by P is no weaker than σ ( U , Y ).The topology generated by P is thus compatible with the duality.Solidity of C and the interchange rule [21, Theorem 14.60] give p ( u ) = sup y ∈ C,y ′ ∈ L { E [ u · y ′ ] | | y ′ | ∗ ≤ | y | ∗ } = sup y ∈ C E [ | u || y | ∗ ] , so p is lower semicontinuous in L and satisfies (A3).By Lemma 21, the set { u · y | y ∈ C } is uniformly integrable so p ( u A ν ) ց A ν ) ∞ ν =1 is a decreasing sequence with P ( A ν ) ց
0. Thus, (A5) holds.This also implies that L ∞ is P -dense in dom p .Any C ∈ C is σ ( L , L ∞ )-compact so an application of bipolar theorem inthe duality pairing ( L , L ∞ ) gives p ◦ ( y ) = inf { γ > | y/γ ∈ C } . Thus dom p ◦ ⊂ Y . As noted earlier, any y ∈ Y belongs to some C ∈ C so Y = ∪ p ∈P dom p ◦ .The σ ( Y , U )-compactness of C ∈ C implies σ C ( u ) < ∞ for any u ∈ U . Thus, U ⊂ ∩ p ∈P dom p . On the other hand, ∩ p ∈P dom p is complete in the P -topology(see Remark 4) so it is complete also in the topology generated by σ ( Y , U )-compact convex sets. Since L ∞ is dense in dom p , we have that U is dense in ∩ p ∈P dom p and thus, U = ∩ p ∈P dom p .Theorem 23 puts us in the setting of Remark 4. Combined with Lemma 19,we thus get the following two results. Corollary 24. If U is τ ( U , Y ) -complete, then it is sequentially σ ( U , Y ) -complete. roof. Let ( u ν ) ∞ ν =1 be a σ ( U , Y )-Cauchy sequence. Since σ ( U , Y ) is strongerthan σ ( L , L ∞ ) which, by [3, Theorem IV.8.6], is sequentially complete, thereexists u ∈ L such that u ν → u in σ ( L , L ∞ ). Since σ ( U , Y )-Cauchy sequencesare bounded in any topology compatible with the pairing, the sequence is alsobounded in the P -topology of Theorem 23. Thus, for any p ∈ P , there exist γ such that p ( u ν ) ≤ γ . Since level-sets of p are closed in L and U = T p , we get u ∈ U . It suffices to show that u ν → u in σ ( U , Y ).By Lemma 19, for any y ∈ Y , ( u ν · y ) ∞ ν =1 is Cauchy in σ ( L , L ∞ ), so bysequential closedness of L again, it converges in σ ( L , L ∞ ) to some ξ ∈ L . ByMazur’s theorem, there is a subsequence of convex combinations ¯ u ν such that¯ u ν → u in L -norm, and thus ¯ u ν · y → u · y in probability. Clearly, ¯ u ν · y → ξ in σ ( L , L ∞ ), so we must have ξ = u · y .When U is τ ( U , Y )-complete, we get the following version of Lemma 21. Corollary 25.
Assume that U is τ ( U , Y ) -complete. A convex set C ⊂ U isrelatively σ ( U , Y ) -compact if and only if, for every y ∈ Y , the set { u · y | u ∈ C } is relatively σ ( L , L ∞ ) -compact in L .Proof. Since continuous images of relatively compact sets are relatively compact,Lemma 19 gives the necessity. For the sufficiency, it suffices, by Theorem 23and Remark 4, to show sequential relative compactness. Let ( u ν ) be a sequencein C . As in the proof of Lemma 21, we get that there is u ∈ L such that, forevery y ∈ Y and ǫ > | E [( u ν − u ) · y ] | ≤ ǫ, for ν large enough, so ( u ν ) is Cauchy in σ ( U , Y ). By Corollary 24, ( u ν ) convergesto u . Appendix
This appendix studies integration of measurable not-necessarily bounded func-tions with respect to a real-valued finitely additive measure m . Define r m : L → R by r m ( η ) := sup u ′ ∈ L ∞ { Z Ω u ′ dm | | u ′ | ≤ η } . Lemma 26.
For any real-valued finitely additive measure m ,1. Relative to L ∞ , r m ( η ) = sup u ′ ∈ L ∞ (cid:26)Z Ω η ( u ′ dm ) (cid:12)(cid:12)(cid:12)(cid:12) | u ′ | ≤ (cid:27) ≤ || η || L ∞ || m || T V . In particular, r m is L ∞ -norm continuous and sublinear relative to L ∞ + .2. For every η ∈ L , r m ( η ) = lim ν ր∞ r m ( η ∧ ν )24 . r m is positively homogeneous and subadditive and r m ( η ′ ) ≤ r m ( η ) when-ever η ′ ≤ η .Proof. The expression in 1 follows from the change of variables ˜ u = ηu ′ . Toprove 2, the inequality r m ( η ) ≥ lim ν r m ( η ∧ ν ) is clear. To prove the oppositeinequality, let α ∈ R with r m ( η ) > α . There exists u ′ ∈ L ∞ with | u ′ | ≤ η and r m ( | u ′ | ) > α . Then | u ′ | ∧ ν → | u ′ | in L ∞ -norm, so monotonicity and 1 givelim r m ( η ∧ ν ) ≥ lim r m ( | u ′ | ∧ ν ) > α. In 3, only subadditivity requires a proof. Given η , η ∈ dom p , we have( η + η ) ∧ ν ≤ η ∧ ν + η ∧ ν . Indeed, a concave function vanishing at theorigin is subadditive on the positive reals. Thus, by 1 and 2, r m ( η + η ) = lim sup ν r m (( η + η ) ∧ ν ) ≤ lim sup ν ( r m ( η ∧ ν ) + r m ( η ∧ ν )) ≤ lim sup ν r m ( η ∧ ν ) + lim sup ν r m ( η ∧ ν )= r m ( η ) + r m ( η ) , which proves the subadditivity.Define ρ m : L → R by ρ m ( u ) := r m ( | u | ) . Theorem 27.
For any real-valued finitely additive measure m ,1. ρ m is symmetric and sublinear, and ρ m ( u ′ ) ≤ ρ m ( u ) whenever | u ′ | ≤ | u | ,2. for any u ∈ dom ρ m and ǫ > , there exists u ′ ∈ L ∞ with ρ m ( u − u ′ ) < ǫ ,3. R Ω udm has a unique ρ m -continuous linear extension from L ∞ to dom ρ m ,4. if m is purely finite additive, there exists a decreasing ( A ν ) ∞ ν =1 ⊂ F with P ( A ν ) ց and R Ω u Ω \ A ν dm = 0 for all u ∈ dom ρ m .Proof. Properties in 1 are clear. To prove 2, assume first that m is nonnegative.Given u i ∈ dom ρ m ∩ L and ǫ >
0, let ˜ u i ∈ L ∞ be such that 0 ≤ ˜ u i ≤ u i and ρ j ( u i ) ≤ h ˜ u i , m i + ǫ . Then ˜ u + ˜ u ≤ u + u and ρ m ( u ) + ρ m ( u ) ≤ h ˜ u + ˜ u , m i + 2 ǫ ≤ ρ m ( u + u ) + 2 ǫ. Since ǫ > ρ m is superlinear on dom ρ m ∩ L . Given u ∈ dom ρ m and ǫ >
0, Lemma 26 gives ρ m ( u + ) ≤ ρ m ( u + ∧ ν ) + ǫ for ν large enough. Bysuperlinearity, ρ m ( u + − u + ∧ ν ) + ρ m ( u + ∧ ν ) ≤ ρ m ( u + ) ≤ ρ m ( u + ∧ ν ) + ǫ. ρ m ( u − − u − ∧ ν ) ≤ ǫ , so ρ m ( u − π ν B u ) ≤ ǫ by sublinearity of ρ m .By [22, Theorem 1.12], general m ∈ M can be written as m = m + − m − fornonnegative m + , m − ∈ M , so ρ m ( u − π ν B u ) ≤ ρ m + ( u − π ν B u ) + ρ m − ( u − π ν B u ) ≤ ǫ for ν large enough.We have R Ω udm ≤ ρ m ( u ) on L ∞ , so, by Hahn-Banach, there exists a ρ m -continuous linear extension of m to dom ρ m . Since L ∞ is dense in dom ρ m , theextension is unique. If m is purely finitely additive, there exists ( A ν ) ∞ ν =1 ⊂ F with P ( A ν ) ց R Ω u Ω \ A ν dm = 0 for all u ∈ L ∞ . Note that r m inheritsthis property so that ρ m and the integral does as well. References [1] Colin Bennett and Robert Sharpley.
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