Topological duals of locally convex function spaces

This paper studies topological duals of locally convex function spaces that are natural generalizations of Fréchet and Banach function spaces. The dual is identified with the direct sum of another function space, a space of purely finitely additive measures and the annihilator of L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document}. This allows for quick proofs of various classical as well as new duality results e.g. in Lebesgue, Musielak–Orlicz, Orlicz–Lorentz space and spaces associated with convex risk measures. Beyond Banach and Fréchet spaces, we obtain completeness and duality results in general paired spaces of random variables.


Introduction
Banach function spaces (BFS) provide a convenient set up for functional analysis in spaces of measurable functions. Many well known properties of e.g. Lebesgue spaces and Orlicz spaces extend to BFS with minor modifications; see e.g. [1,20,22,38]. Extensions to Fréchet function spaces have been studied e.g. in [4]. This paper studies topological duals of more general locally convex function spaces where the topology is generated by an arbitrary collection of seminorms satisfying the usual BFS axioms.
Building on the classical result of Yosida and Hewitt [37,Section 2] on the dual of L ∞ , we identify the topological dual as the direct sum of another space of random variables (Köthe dual), a space of purely finitely additive measures and the annihilator of L ∞ . The last two components have a singularity property that has been found useful, e.g., in the analysis of convex integral functionals by Rockafellar [32] in the case of L ∞ and by Kozek [19] in the case of Orlicz spaces. In the case of L ∞ , the last component in the dual vanishes while in other Orlicz spaces, the second one vanishes; see [28,Chapter IV]. Our result thus unifies the two seemingly complementary cases.
The main result is illustrated first by simple derivations of various existing as well as new duality results in Musielak-Orlicz, Marcinkiewicz, Lorentz and Orlicz-Lorentz spaces. We go beyond the existing BFS settings by identifying topological duals of the space of random variables with finite moments, generalized Musielak-Orlicz spaces as well as spaces of random variables associated with convex risk measures. The last instance has attracted attention in the recent literature of mathematical finance and insurance; see e.g. [21,27] and [11].
Section 5 is concerned with dual pairs of linear spaces of random variables that play a central role e.g. in stochastic optimization and optimal control; see e.g. [36] and the examples in [33]. We show that the corresponding Mackey and strong topologies are generated by (uncountable) collections of seminorms satisfying the usual BFS axioms. We then obtain duality and completeness results as corollaries of the main results of the paper.
The rest of the paper is organized as follows. Section 2 reviews the duality theory for L ∞ . Section 3 extends the notion of an integral with respect to a finitely additive measure to measurable not necessarily bounded random variables. Section 4 defines a general locally convex space of random variables and gives the main result of the paper by characterizing the topological dual of a space. Section 5 studies spaces of random variables in separating duality. Section 6 applies the main result to characterize the topological dual in various known and new settings.

Topological dual of L ∞
Let ( , F, P) be a probability space with a σ -algebra F and a countably additive probability measure P. This section reviews the topological dual of the Banach space L ∞ of equivalence classes of essentially bounded measurable functions on a probability space ( , F, P). We consider R n -valued functions and endow L ∞ with the norm 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 -valued measures on ( , F) and let M s be set of those m ∈ M which are singular ("purely finitely additive" in the terminology of [37]; see [37,Theorem 1.22]) in the sense that there is a decreasing sequence (A ν ) ∞ ν=1 ⊂ F with P(A ν ) 0 and |m| * ( \ A ν ) = 0. Given m ∈ M, the set function |m| * : F → R is defined by where ith components of m + ∈ M and m − ∈ M are the positive and negative parts, respectively, of the ith component m i of m; see [37,Theorem 1.12].
Recall that the space E of R n -valued simple random variables (i.e. piecewise constant with a finite range) is dense in L ∞ . Given m ∈ M, the integral of a u ∈ E is defined by where A j ∈ F and α j ∈ R n , j = 1, . . . , m are such that u = m j=1 α j 1 A j On L ∞ , the integral is defined as the unique norm continuous linear extension from E to L ∞ .
The following is from [37,Theorem 2.3] except that we do not assume that the underlying measure space is complete; see also [2,Sections 4.7 and 10.2]. The proof uses [9,Theorem 20.35] which does not rely on the completeness but identifies the dual of L ∞ with the space of finitely additive measures that are absolutely continuous with respect to P. Results of [37,Section 1] on decomposition of finitely additive measures then complete the proof. The above are concerned with real-valued random variables but the extension to the vector-valued case is straightforward; see [35,Lemma 1] for an extension to Banach space-valued random variables. Throughout this paper, the expectation of a random variable z ∈ L 1 is denoted by E [z]. The inner product of two vectors ξ, η ∈ R n is denoted by ξ · η.
Theorem 1 (Yosida-Hewitt) The topological dual (L ∞ ) * of L ∞ can be identified with M in the sense that for every u * ∈ (L ∞ ) * there exists a unique m ∈ M such that where the integral is defined componentwise. The dual norm is given by Moreover, M = L 1 ⊕ M s in the sense that for every m ∈ M there exist unique y ∈ L 1 and m s ∈ M s such that We have m s = 0 if and only if u1 A ν , u * → 0 for every u ∈ L ∞ and every decreasing (A ν ) ∞ ν=1 ⊂ F such that P(A ν ) 0. Proof Assume first that n = 1. By [9,Theorem 20.35], the dual of L ∞ can be identified with the linear space of finitely additive P-absolutely continuous measures m in the sense that every u * ∈ (L ∞ ) * can be expressed as u, u * = udm and, conversely, any such integral belongs to (L ∞ ) * . By [37,Theorem 1.24], there is a unique decomposition m = m a + m s , where m a is countably additive and m s is purely finitely additive. The construction in [37] also shows that m a and m s are absolutely continuous with respect to m and thus, absolutely continuous with respect to P as well. By [37,Theorem 1.22], there is a decreasing sequence (A ν ) ∞ ν=1 ⊂ F such that P(A ν ) 0 and m s ( \ A ν ) = 0. The functional y s ∈ (L ∞ ) * given by u, y s := udm s then has the property in the statement. By Radon-Nikodym, there exists a y ∈ L 1 such that To prove the last claim, it is clear that the given condition holds if m s = 0. To prove the converse, let u * ∈ (L ∞ ) * and consider the representation in terms of y ∈ L 1 and m s ∈ M s given by the second claim. Let A ν be the sets in the characterization of the singularity of m s . By [37,Theorems 1.12 and 1.17], m s = m s+ − m s− for nonnegative purely finitely additive m s+ and m s− . Given > 0, [37,Theorem 1.21] gives the existence of A ∈ F such that m s+ ( \ A) < and m s− (A) < . We have Under the given condition, the left side converges to zero. Since > 0 was arbitrary, m s+ = 0. By symmetry, m s− = 0 so m s = 0.
By [37,Theorem 2.3], the dual norm of · L ∞ is given by m T V := m + ( ) + m − ( ). This completes the proof of the case n = 1. The general case follows from the fact that the dual of a Cartesian product of Banach spaces is the Cartesian product of the dual spaces with the norm which completes the proof.

Extension of the integral
In [37] and in Sect. 2, integrals with respect to an m ∈ M were defined only for elements of L ∞ as norm-continuous extensions of integrals of simple functions. Weakening the topology, it is possible to extend the definition of the integral to a larger space of measurable functions using Daniell's construction much as in [3,Chapter II] which considered countably additive integrals of arbitrary (not necessarily F-measurable) functions.
Another approach to integration of unbounded functions with respect to finitely additive measures is that of Dunford; see Dunford and Schwartz [6] or Luxemburg [23].
A benefit of the Daniell extension adopted here is that it gives rise to a simpler definition of integrability that is easier to verify for larger classes of measurable functions.
Given m ∈ M, we define ρ m : L 0 → R by We denote The extension of the one-dimensional integral in Theorem 29 gives the following.
Proof The extension is given by where the integrals on the right are the extensions of the one-dimensional integrals given in Theorem 29. We get where the second inequality comes from Theorem 29. The sets A ν can be taken as the unions of the componentwise sets given by Theorem 29.
We call the extension in Theorem 2 the m-integral of u and denote it by udm.
The elements of dom ρ m will be said to be m-integrable. If m is countably additive, then, by e.g. [34,Theorem 14.60], where y is the density of m, and thus, In this case, the integral is the Lebesgue integral.

Topological duals of spaces of random variables
This section contains the main result of the paper. The setup extends that of Banach function spaces by replacing the norm by an arbitrary collection of seminorms thus covering more general locally convex spaces of random variables. The main result identifies the topological dual of the space with the direct sum of a space of random variables and two spaces of singular functionals, the first of which is represented by finitely additive measures while the second is the orthogonal complement of L ∞ . Let L 0 be the linear space of R n -valued random variables. Let P be a collection of sublinear (i.e. convex and positively homogeneous) functions p : L 0 → R with p(u) = p(−u) for all u ∈ L 0 , define and endow L P with the locally convex topology generated by P. Recall that dom p := {u ∈ L 0 | p(u) < ∞}. Our aim is to characterize the topological dual L * P of L P . To this end, we will assume that, for each p ∈ P, surely.
Occasionally, we will also assume the following (A3) p(u1 A ν ) 0 for all u ∈ L ∞ and decreasing sequence ( It is clear that (A3) and (A4) are implied by the following p(u ν ) 0 for all (u ν ) ∈ L ∞ such that |u ν | 0, p(u ν ) 0 for all (u ν ) ∈ L P such that |u ν | 0, respectively. The case where P is a singleton has been extensively studied ever since the publication of [22]; see e.g. the monographs [1,20,38,39]. When P is a singleton satisfying merely (A2), L P is usually called a Banach function space provided it is complete. A sufficient condition for completeness in the general case is given in Remark 9 below.
Our approach relies on (A1) which is assumed also e.g. in [1] but avoided in many other treatments of Banach function spaces. The assumption allows for simple proofs building on Theorem 1 and it provides more information on the structure of the topological dual; see Theorem 6 below. Under a mild condition, spaces satisfying (A2) alone are isomorphic to spaces satisfying both (A1) and (A2).

Remark 3
LetP be a collection of seminorms satisfying (A2) but not necessarily (A1). If there existsū ∈ LP such thatū j > 0 almost surely for all j, then LP is linearly isomorphic to a space L P satisfying (A1) and (A2). The elements of L * P can thus be expressed asũ where i is the isomorphism and u * ∈ L * P . Proof Define i : L 0 → L 0 by i(u) j := u j /ū j and, for everyp ∈P, let p(u) := p(i −1 (u)). By (A2), so P := {p • i −1 |p ∈P} satisfies (A1) and (A2). The restriction of A to LP is an isomorphism between LP and L P .
A seminorm p is said to be rearrangement invariant (or symmetric) if p(u) = p(ũ) whenever u andũ have the same distribution in the sense that P(|u| > τ) = P(|ũ| > τ ) for all τ ≥ 0.

Remark 4
Consider the scalar case n = 1. Given p ∈ P, let Under (A2), condition (A3) is equivalent to lim t 0φ p (t) = 0. If lim t 0φ p (t) > 0, then dom p = L ∞ . If p is rearrangement invariant, then, for any A ∈ F with P(A) = t,φ where the common value is known as the fundamental function; see e.g. [1]. In this case, L P = L ∞ unless (A3) holds.
For the converse, let u ∈ L ∞ and (A ν ) ∞ ν=1 ⊂ F with t ν := P(A ν ) 0. By (A2), If inf t>0φ p (t) > 0, then For each p ∈ P, we define a sublinear symmetric function p • on the space M of P-absolutely continuous finitely additive measures by The following shows, in particular, that every m ∈ dom p • decomposes uniquely into the sum of countably additive and singular measures both in dom p • . Lemma 5 Assume that p ∈ P satisfies (A2) and let m ∈ dom p • . Every u ∈ dom p is m-integrable and There exist unique y ∈ L 1 ∩ dom p • and m s ∈ M s ∩ dom p • such that Given m s ∈ M s ∩dom p • , there exists a decreasing (A ν ) ∞ ν=1 ⊂ F such that P(A ν ) 0 and u1 \A ν dm s = 0 ∀u ∈ dom p.
and the existence of the sequence (A ν ) ∞ ν=1 for very m s ∈ M s ∩ dom p • . By Theorem 1, there exist unique y ∈ L 1 and m s ∈ (L ∞ ) s such that m = y P +m s . Let α < p • (y) and α s < p • (m s ) and u, u s ∈ L ∞ such that p(u), p(u s ) ≤ 1 and uyd P ≥ α and u s dm s ≥ α s .
where λ ∈ (0, 1). By convexity and (A2), 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, so, by the last claim of Theorem 1, condition (A3) implies m s = 0. Let the set of P-absolutely continuous finitely additive measures m ∈ M such that p • (m) < ∞ for some p ∈ P. The set of purely finitely additive elements of M P • will be denoted by M P • s . The set of densities y = dm/d P of countably additive m ∈ M P • will be denoted by L P • . By Lemma 5, M P • = L P • ⊕ M P • s . In the setting of Banach function spaces where P is a singleton, L P • is often called the "associate space" or the Köthe dual of L P ; see e.g. [1,8,39] and Lemma 8 below.
The following is the main result of this section. It identifies the topological dual L * P of L P with the direct sum of M P • = L P • ⊕ M P • s and the annihilator

Theorem 6
We have For every u ∈ L P and m ∈ M P • , In particular, restricted to M P • , p • coincides with the polar seminorm of p, i.e.
Given w ∈ (L ∞ ) ⊥ and u ∈ L P , there exists a decreasing sequence To prove the opposite inclusion, let u * ∈ L * P . There exists p ∈ P and γ > 0 such that u * ≤ γ p. Assumption (A1) implies that u * is continuous in L ∞ . By Theorem 1, there exists a unique m ∈ M such that u, u * = udm for all u ∈ L ∞ . Since u * ≤ γ p, we have m ∈ dom p • , so m is continuous on L P by Lemma 5. Now w := u * − m belongs to (L ∞ ) ⊥ , so u * has the required decomposition. Given another decomposi- Thus ud(m −m) = 0 for all u ∈ L ∞ , so m −m = 0 and hence also w −w = 0, so the decomposition is unique.
The inequality follows directly from that of Lemma 5. Let u ∈ L P and A ν : When P is a singleton, we are in the setting of [38,Theorem 15.70.2], where L * P is decomposed into the direct sum of L P • and "singular elements". Theorem 6 gives a more precise description of the singular elements as the direct sum of M P • s and (L ∞ ) ⊥ . Applications will be given in the following sections.
Let M P be the closure of L ∞ in L P .

Corollary 7 We have
Proof By Hahn-Banach, a continuous linear functional on M P is a restriction to M P of a continuous linear functional on L P . The first two claims thus follow from Theorem 6.
To prove the last claim, take any u ∈ L P and define u ν ∈ L ∞ as the pointwise projection of u on the Euclidean ball of radius ν.
The we end this section by giving some basic properties of L P • . The Köthe dual of L P is the linear space By definition, L P • ⊆ L P . Lemma 8 below gives sufficient conditions for the converse. Recall that a locally convex space is barreled if every closed convex absorbing set is a neighborhood of the origin. By the Baire category theorem, Banach and Fréchet spaces are barreled.
If the P-topology is barreled and stronger than that of L 0 , then L P • = L P .
so 2 holds. The inequality in 3 follows from the inequality in Lemma 5.
To prove the last claim, let y ∈ L 0 be such that E[u · y] < ∞ for all u ∈ L P . Then p y (u) := E[|u||y|] < ∞ for all u ∈ L P . By Fatou's lemma, p y is lsc in the L 0 -topology. By assumption, p y is lsc also in the P-topology. When the P-topology is barreled, p y is continuous (see e.g. [33,Corollary 8B]), so y → E[u · y] is continuous as well. By Theorem 6, y ∈ L P • .
In the setting of Banach function spaces where P is a singleton, the last claim of Lemma 8 recovers [1,Lemma 1.2.6]. The following gives sufficient conditions for the space L P to be complete and thus, barreled, when the topology is metrizable.

Remark 9
If the P-topology is stronger than that of L 0 and p ∈ P are lower semicontinuous on L 0 , then L P is complete. In this case, L P is a Banach/Fréchet (and, in particular, barreled) if P is a singleton/countable.
If p(u) = ρ(|u|) for an nondecreasing ρ : L 0 → R, the function p is lsc in probability if and only if ρ has the Fatou property: for any sequence Proof If (u ν ) is a Cauchy net in L P , it is Cauchy also in L 0 so, by completeness of L 0 , it L 0 -converges to an u ∈ L 0 . Being Cauchy in L P means that for every > 0 and p ∈ P, there is aν such that The lower semicontinuity then gives so u ∈ L P , by triangle inequality, and (u ν ) converges in L P to u. Thus L P is complete.

Solid spaces of random variables
Axiom (A1) implies that L P contains L ∞ while axiom (A2) implies that it is solid in the sense that it contains every u ∈ L 0 for which there exists u ∈ L P with |u j | ≤ |u j | for all j = 1, . . . , n. By Lemma 8, L P • is solid as well. This section starts with an arbitrary pair (U, Y) of solid spaces of random variables in separating duality under the bilinear form We assume that both U and Y contain L ∞ and show that the Mackey and the strong topologies arise from (uncountable) collections of seminorms on L 0 satisfying the axioms of Sect. 4. We then obtain completeness and duality results as corollaries of the results there.
The weak topology generated by Y on U will be denoted by σ (U, Y). Similarly on Y. The Mackey topology τ (U, Y) on U is generated by the collection of seminorms defined as the support functions By the bipolar theorem, this is the topology generated by all τ (U, Y)-lower semicontinuous seminorms on U. By the Mackey-Arens theorem, the Mackey topology on U is the finest topology under which the topological dual of U coincides with Y. Since compact sets are bounded, the strong topology is stronger than τ (U, Y).
Lemma 11 below does not require solidity but merely decomposability in the sense that u1 A +ū1 \A ∈ U for every u ∈ U,ū ∈ L ∞ and A ∈ F. Example 10 Solid spaces containing L ∞ are decomposable but there are decomposable spaces that are not solid. Indeed, let = [0, 1], F the Borel sigma algebra and P the Lebesgue measure. Let u(ω) := ω − 1 4 + ω − 1 2 and U := L ∞ + L, where L is the linear span of functions of the form u1 A with A ∈ F. Then U is decomposable, by construction, but not solid, since it does not containū(ω) = ω − 1 4 while 0 <ū < u.
The following is Lemma 6 from [26].

Lemma 11
If U and Y are decomposable, then L ∞ ⊆ U ⊆ L 1 and

Lemma 12
If U is solid, then, for every u ∈ U, Lemma 12 implies, in particular, that axiom (A3) is necessary for the second claim of Corollary 7.

Corollary 13 In the setting of Corollary 7, (A3) holds if and only if M
Proof By Corollary 7, (A3) implies M * P = L P • . On the other hand, if M * P = L P • , the topology of M P cannot be stronger than τ (M P , L P • ). In that case, Lemma 12 The following characterization of σ (U, Y)-compact sets will be useful. In the case of Orlicz spaces, a similar characterization of relative compactness can be found e.g. in [5,28].

Lemma 14
Given C ⊂ U, the following are equivalent.
Proof Since continuous images of precompact sets are precompact, 3 implies 2, and, by Lemma 12, 1 implies 3. Clearly, 2 implies 3, so it suffices to show that 2 and 3 imply 1. Let (u ν ) be a net in C. Since Y contains constants, the sets {u j | u ∈ C} are σ (L 1 , L ∞ )precompact by 3. Thus there is a subnet and u ∈ C such that u ν → u in σ (L 1 , L ∞ ). Let y ∈ Y and > 0. By the Dunford-Pettis theorem, 2 implies that {u · y | u ∈ C} is The solid hull s(C) of a set C ⊆ U is the smallest solid set containing C. Clearly,

Corollary 15
The solid hull of a σ (U, Y)-bounded set is σ (U, Y)-bounded and the solid hull of a σ (U, Y)-precompact set is σ (U, Y)-precompact.

Proof
We have E[u · y] < ∞ if and only if E j |u j ||y j | < ∞, which implies the first claim. By Lemma 14 and the Dunford-Pettis theorem, a set C ⊆ U is σ (U, Y)precompact if and only if C y, j := {u j y j | u ∈ C} is uniformly integrable for every y ∈ Y and j = 1, . . . , n. Uniform integrability of C y, j means that, for every > 0, there exists M > 0 such that E|1 |ū j y j |≥Mū j y j | < for everyū ∈ C. Clearly, uniform integrability of C y, j implies that of s(C) y, j . Thus, if C is precompact, then s(C) is precompact.
Let C s be the collection of solid hulls of σ (Y, U)-bounded sets. We define P s as the collection of functions p C on L 0 defined by where C ∈ C s and the expectation is defined as +∞ unless the positive part of u · y is integrable. Analogously, we define C τ as the collection of solid hulls of σ (Y, U)compact sets and P τ as the collection of functions p C with C ∈ C τ . By Corollary 15, the restrictions of P s and P τ to U generate the strong and the Mackey topologies, respectively. Note that solid hulls of convex sets in U need not be convex. For subsets of L 0 + , however, taking the solid hull and convex hull commute; see [18,Proposition 1.1].

Lemma 16
The members of P s satisfy (A1) and (A2) while the members of P τ satisfy (A1)-(A4). Both P s and P τ contain the L 1 -norm and their members are L 0 -lsc.
Proof Since σ (Y, U)-bounded sets are L 1 -bounded, the functions p C are dominated by the L ∞ -norm. Thus, P s satisfies (A1). Since the sets C ⊂ C s are solid, [34,Theorem 14.60] gives |u j ||y j |, so p C satisfies (A2). By Fatou's lemma, each supremand is L 0 -lsc so p C is L 0 -lsc as well.
Since P τ ⊂ P s , axioms (A1) and (A2) are again satisfied by P τ and its elements are L 0 -lsc. Given C ∈ C τ and u ∈ dom p C , Lemma 14 and the Dunford-Pettis theorem imply that the set {u · y | y ∈ C} is uniformly integrable so p C (u1 A ν ) 0 whenever (A ν ) ∞ ν=1 is a decreasing sequence with P(A ν ) 0. Thus, P τ satisfies (A4). By Banach-Alaoglu, the unit ball B of L ∞ is σ (L ∞ , L 1 )-compact so, by Lemma 11, it is σ (Y, U)-compact as well. Thus, B ∈ C τ , so P τ contains the L 1 -norm and thus P s does as well.
Recall that the Köthe dual of a space U of measurable functions is the linear space Our assumptions on U and Y imply that they are contained in each other's Köthe duals. The following shows, in particular, that if U is equal to the Köthe dual of Y, then U and Y arise from the construction of Sect. 4 with the Mackey-seminorms P τ .

Theorem 17
We have Y = L P • τ , U ⊂ L P τ ⊂ Y and the following are equivalent For every y ∈ Y, there is a p ∈ P τ such that p • (y) < ∞ so, by the Hölder's inequality in Lemma 8, This proves the second claim.
By the second claim, 1 implies 2. Lemma 16 and Remark 9 imply that L P τ is complete. Thus, 2 implies 3. On the other hand, by Lemma 16, P τ satisfies (A1)-(A4) so L ∞ is dense in L P τ . Since U is decomposable, it contains L ∞ . Thus, if U is complete in the relative topology of P τ , it has to coincide with L P τ . Thus, 3 implies 2.
We next show that 4 implies 1. Let u ∈ L 1 be in the Köthe dual and let u ν ∈ L ∞ be the pointwise projection of u to the Euclidean ball with radius ν. By dominated convergence, E[u ν · y] → E[u · y] for all y ∈ Y. Thus, (u ν ) ∞ ν=1 is weakly Cauchy so 4 implies that it has a σ (U, Y)-limit u ∈ U. It follows that E[u · y] = E[u · y] for all y ∈ Y so u = u .
It remains to show that 2 implies 4. Let (u ν ) ∞ ν=1 be a σ (U, Y)-Cauchy sequence. Since σ (U, Y) is stronger than σ (L 1 , L ∞ ) which, by [6,Theorem IV.8.6], is sequentially complete, there exists u ∈ L 1 such that u ν → u in σ (L 1 , L ∞ ). Since σ (U, Y)-Cauchy sequences are bounded in any topology compatible with the pairing, the sequence is also bounded in the P τ -topology. Thus, for any p ∈ P τ , there exists γ ∈ R such that p(u ν ) ≤ γ . Since level-sets of p are closed in L 1 and U = L P τ , we get u ∈ U. It suffices to show that u ν → u in σ (U, Y).
The following shows, in particular, that if U and Y are Köthe duals of each other, then they arise from the construction of Sect. 4 with the strong seminorms P s . Theorem 6 then yields a characterization of the strong dual of U.
Proof Since U ⊂ L P s ⊂ L P τ , the first claim follows from Theorem 17. Since L P • τ ⊆ L P • s , Theorem 17 implies Y ⊆ L P • s . On the other hand, since U ⊆ L P s , the Hölder's inequality in Lemma 8 implies When U = L P s and Y = L P • s , we are in the setting of Sect. 4. By Lemma 16, P s satisfies (A1) and (A2), so the last claim follows from Theorem 6.
In the setting of Theorem 6, one may wonder what is the strong topology generated by L P • on L P .

Theorem 19
If L P is barreled and p ∈ P are σ (L P , L P • )-lsc, then the strong topology generated by L P • on L P coincides with the P-topology.
Proof If p is σ (L P , L P • )-lsc, Theorem 6 and the bipolar theorem imply that It follows that the level sets of the functions p • generate the L P -topology. Since the level sets are bounded, the L P -topology is weaker than the strong topology generated by L P • . On the other hand, if L P is barreled, then the elements of P s are L P -continuous.

Applications
This section applies the results of the previous sections to more specific situations. We obtain quick proofs of many well known as well as new results.

Random variables with finite moments
Given an increasing sequence S ⊂ [1, ∞), let The L p -norms with p < ∞ satisfy (A1)-(A4). The following example is thus a direct consequence of Corollary 7.

Marcinkiewicz and Lorentz spaces
Given a random variable u ∈ L 0 , we will denote the distribution function of |u| by n u (τ ) := P(|u| > τ) and its quantile function by q u (t) := inf{τ ∈ R | n u (τ ) ≤ t}.
In the terminology of Banach function spaces, the quantile function is usually called the "decreasing rearrangement of u; see e.g. [1]. Given a nonnegative concave increasing function φ on [0, 1] with φ(0) = 0, the associated Marcinkiewicz space is the set M φ of u ∈ L 0 with Recall that a probability space is resonant if it is atomless or completely atomic with all atoms having equal measure.
The closure M 0 φ of L ∞ in M φ can be expressed as The topological dual of M 0 φ is and the topological dual of φ is M φ .
Proof We apply Theorem 6 with P = {p} where p(u) = u φ . Since we have M φ ⊂ L 1 and its topology is stronger than the L 0 -topology. By Lemma 30, is the infimal projection of a sublinear function of s and u and thus, sublinear in u. It is also continuous in L 1 . It follows that · φ is sublinear, symmetric and lsc in L 1 .
By Remark 9, M φ is Banach. Since q u ≤ u L ∞ , we have where sup t∈(0,1] t φ(t) < ∞ since φ is concave and strictly positive for t > 0. Thus, (A1) holds. Property (A2) is clear. Given A ∈ F, where the second equality follows from [ If u ∈ L ∞ , q u is bounded, so where the second last inequality follows from concavity of φ. Thus, M 0 φ is closed in M φ so M 0 φ contains the closure of L ∞ . To prove the converse, let u ∈ M 0 φ and u ν = Since u ∈ M 0 φ , this converges to 0 as ν → ∞. Thus, M 0 φ is the closure of L ∞ in M φ . To prove the last claim, we apply Theorem 6 to p • . By Lemma 8, the Lorentz seminorm satisfies (A1) and (A2). If y ν 0 with y ν * φ < ∞, we have q y ν 0, so by monotone convergence, y ν * φ 0. Thus, the Lorenz norm satisfies (A4) as well. The fact that the topological dual of φ is M φ now follows from Theorem 6 and the fact that, by the bipolar theorem, p is the polar of p • .

Modular spaces and Luxemburg norms
This section studies a general class of Banach spaces that arise from a positive symmetric convex function (a convex modular in the terminology of [25]) on L 0 much like Orlicz spaces arise from the Luxemburg norm associated with a given Young functional; see Sect. 6.4 below. Theorem 22 below allows for quick proofs and various extensions of existing results in the theory of Banach function spaces.
Given a set C in a linear space, we will use the notation where the infimum is attained. Moreover, By (H1), p is finite on L ∞ . Since p is lsc on L 0 , it is lsc on L ∞ . Thus, by [33,Corollary 8B], p is continuous in L ∞ and thus (A1) holds. Assumption (A2) is clear from (H2).

The dual of the closure M H of L ∞ in L H can be identified with L H
Let u ν → u in L 0 be such that p(u ν ) ≤ α. This means that H (u ν /α) ≤ 1, so the L 0 -lower semicontinuity of H implies that of p. Let O be an L 0 -neighborhood of the origin. By the boundedness assumption in (H1), there exists λ > 0 such that in L H , so the topology of L H is no weaker than the relative L 0 -topology. By Remark 9, L H is Banach. By Lemma 8, L H * is the Köthe dual of L H .
Let m ∈ M. Since the infimum in the definition of the p is attained, Lagrangian duality (see e.g. [33,Example 1"]) gives where the infimum is attained. It follows that dom p • = pos dom H * . The first claim thus follows from Theorem 6.
The associated Musielak-Orlicz space is the normed space where the infimum is attained. Moreover,

The dual of the closure M of L ∞ in L is
If we have E * (η) ≤ 1. Since sup a (a) > 0 almost surely, η > 0 almost surely. By Fenchel's inequality, Thus, the left is side is bounded in probability, since the right side is so.
If (a) ∈ L 1 for all a > 0, then L ∞ ⊂ dom H and (H3) and (H4) hold by monotone convergence theorem. If dom E is a cone, then (a, ·) ∈ L 1 for all a > 0. All the claims except for the last one thus follow from Theorem 22. Assume the 2 -condition and let Since dom E is a convex set, this implies that it is a cone.

If
is nonrandom, we recover the classical Orlicz spaces and the last part of Theorem 23 implies that, if is finite, then M * s = {0}, while otherwise, L = L ∞ so (L ∞ ) ⊥ = {0}. Extensions to Banach space-valued functions have been studied in [12]. In [25], the assumption (a, ·) ∈ L 1 for all a > 0 is called "local integrability". Thus we recover [25,Theorem 13.17] for probability spaces without assuming local integrability of * ; see also [24,Theorem 2.4.4]. Our characterization of the dual without local integrability seems new. satisfies the first assumption of Theorem 23. The dual of L˜ is thus isomorphic to that of L characterized in Theorem 23. Indeed, the isomorphism is (Aũ)(ω) =ũ(ω)/ρ(ω) so the elements of (L˜ ) * can be expressed as where u * ∈ (L ) * .

Generalized Musielak-Orlicz spaces
Let r be an lsc norm on L 0 satisfying (A1) and (A2) such that the r -topology is stronger than that of L 0 . By Remark 9, the space L r is Banach. Let be as in Sect. 6.4 and define Note that, if r is the L 1 -norm, then L ,r is the Musielak-Orlicz space studied in Sect. 6.4. If is nonrandom and r is the Lorentz-norm The second expression above comes from [31,Theorem 13.3]. It involves the recession function of the conjugate * defined as

Theorem 26
Assume that (|u|) ∈ L r for all u ∈ L ∞ and that {x ∈ L 0 | r (x) ≤ 1} is bounded in L 0 . Endowed with the norm u ,r , the space L ,r is a Banach, its dual may be identified with and L is the Köthe dual of L ,r . For any m ∈ M , the dual norm can be expressed as where the infimum is attained. Moreover, The Given an L 0 -converging sequence u ν → u, the pointwise lower semicontinuity of gives lim inf (u ν ) ≥ (u) so the lower semicontinuity and (A2) of r give so H is L 0 -lsc. As in the proof of Theorem 23, there exists an η ∈ L 0 strictly positive such that (|u|) ≥ |u|η − * (η) almost surely. We have where the right side is L 0 -bounded since {ξ ∈ L 0 | r (ξ ) ≤ 1} is L 0 -bounded by assumption. Since (u) ∈ L r for all u ∈ L ∞ , we have L ∞ ⊂ dom H . Thus, H satisfies (H1)-(H2). We compute the conjugate H by employing conjugate duality; see [33]. The function is convex and increasing in the partial order of L 0 so the function F(x, u) where, by positive homogeneity of r , For any u ∈ L ∞ , the function F(·, u) is continuous on L r , so [33,Theorem 17] gives where the infimum is attained. Thus, by Theorem 22 and positive homogeneity of , The dual Luxemburg norm can be expressed as Since is positively homogeneous, we have pos dom H * = {m ∈ M | ∃x * ∈ L * r : (x * , m) < ∞}.
The proof of Theorem 26 gives also the expression and the infimum is attained. Under condition (b) in the theorem, we have M s = {0} and If r is the Lorentz-norm associated with a concave increasing function φ, then, by Theorem 21, r • is the Marcinkiewicz-norm so The above characterization of the Köthe dual thus extends that in [16,Corollary 4.12] and [15,Theorem 2.2] to random in the case of a finite underlying measure. The singular components of the dual have been analyzed in the recent article [14].
The fact that (L ρ ) * = L α ⊕ (L ∞ ) ⊥ under the Lebesgue property sharpens [21,Theorem 4.12] which states that each u * ∈ (L ρ ) * can be expressed uniquely as u * = y + u s for some y ∈ L 1 and u s ∈ (L ∞ ) ⊥ . The other statements seem new.
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Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 1. For any η ∈ L ∞ + , In particular, r m is L ∞ -norm continuous and sublinear relative to L ∞ + . 2. For every η ∈ L 0 + , 3. r m is subadditive in the sense that r m (η 1 + η 2 ) ≤ r m (η 1 ) + r m (η 2 ) ∀η 1 , η 2 ∈ dom r m .
We have udm ≤ ρ m (u) on L ∞ , so, by Hahn-Banach, there exists a ρ mcontinuous linear extension of m to dom ρ m . Since L ∞ is dense in dom ρ m , the extension is unique. If m is purely finitely additive, there exists (A ν ) ∞ ν=1 ⊂ F with P(A ν ) 0 and u1 \A ν dm = 0 for all u ∈ L ∞ . Note that r m inherits this property so that ρ m and the integral does as well.
The following was used in the proof of Theorem 21. Its proof was given as an exercise on page 89 of [1]. The following lemma was used in the proof of Proposition 24. Let : R × → R be a convex normal integrand in the sense that ω → {(ξ, α) | (ξ, ω) ≤ α} is a convex-valued measurable mapping (see [34,Chapter 14]). The associated integral functional E : L 0 → R is defined by