INTEGRATION with VECTOR VALUED MEASURES M. M. Rao

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2013.33.5429 DYNAMICAL SYSTEMS Volume 33, Number 11&12, November & December 2013 pp. 5429–5440

INTEGRATION WITH VECTOR VALUED MEASURES

M. M. Rao

Unversity of California, Riverside Riverside, CA 92521, USA

Dedicated to Jerome A. Goldstein on the occasion of his 70th birthday

ABSTRACT. Of the many variations of vector measures, the Fréchet variation is finite valued but only subadditive. Finding a ‘controlling’ finite measure for these in several cases, it is possible to develop a useful integration of the Bartle-Dunford-Schwartz type for many linear metric spaces. These include the generalized Orlicz spaces, Lφ(µ), where φ is a concave φ-function with applications to stochastic measures Z(·) into various Fréchet spaces useful in prediction theory. In particular, certain p-stable random measures and a (sub) class of these leading to positive infinitely divisible ones are detailed.

1. Introduction. If (S, S, µ) is a measure space where S is a δ(or σ)-ring of subsets of a space S, and (X , ∥ · ∥) is a linear metric space, then a mapping Z : S → X is termed a vector valued measure (or simply a vector measure) if it is σ-additive in the sense that for any disjoint sequence {An, n ≥ 1} ⊂ S whose union is in S, then one has: ∑∞ ∪∞ Z( n=1An) = Z(An), (1) n=1 the series on the right converging in the metric topology of X . Note that by a classical result, originally due to S. Kakutani for general metric groups and extended by V. L. Klee for abelian groups admitting a metric under which it is complete having an invariant metric, one can always consider the analysis for functionals to be (metric) translation invariant on X . Such a linear space is termed a Fr´echet space hereafter. (See Rolewicz∑ (1972) for details.) If f : S → R(C) is a simple n function, represented as f = i=1 aiχAi , then deﬁne as usual an ‘integral’:

∫ ∑n fdZ = aiZ(Ai ∩ A), (∈ X ,A ∈ S) (2) A i=1 where Ai ∈ S, disjoint. Observe that the left side is uniquely deﬁned (does not depend on the representation of f above) and so (2) is unambiguous. An extension of it for a more general f which is a pointwise limit of simple functions (by the structure theorem of measurable functions for S) with the σ-additivity of Z, is desired. This means if fn, gn, n ≥ ∫1 are such that fn → f ← gn, n → ∞ so that − → − → → ∞ fn gn 0, both pointwise, then A(fn gn)dZ 0 as n must hold under standard conditions. If X is not ﬁnite dimensional there is no ‘Lebesgue type’ limit

2010 Mathematics Subject Classification. Primary: 46G10, 60H05; Secondary: 28B05, 28C20. Key words and phrases. Vector measures in Fréchet spaces, controling measures, independently valued measures, stochastic measures in generalized Orlicz spaces, prediction theory. 5429 5430 M. M. RAO theorem to guarantee this, but one needs it to extend the work for all such functions. This is solved by Bartle-Dunford-Schwartz when (X , ∥ · ∥) is a Banach space who showed that there exists a ‘controlling’ finite measure λ for Z on S, implying that Z ≪ λ using the fact that X ∗ separates points of X . But this result on existence of such a λ needs a substitute in the Fréchet case, covering X = Lp(µ), p ≥ 0. The Vitali and Fréchet variations are now recalled in the following: Definition 1. If Z :Σ → X is a vector measure, then the Vitali variation, denoted |Z|(·) is given by (∥ · ∥ being the metric in X ):

∑n |Z|(A) = sup{ ∥Z(Ai)∥ : Ai ∈ S(A), disjoint}, (3) i=1 where S(A) = {A ∩ B ∈ S, A, B ∈ S}, is the trace of S on A. The Fr´echet variation ∥Z∥ is introduced, with Ai disjoint again, by:

∑n ∥Z∥(A) = sup{∥ aiZ(Ai)∥ : ai ∈ C, |ai| ≤ 1,Ai ∈ S(A)}. (4) i=1

Here on the right ∥ · ∥ stands for the norm or metric of X respectively. It is clear that |Z|(A) ≥ ∥Z∥(A), and if X is the space of scalars then it is well-known that |µ|(A) ≤ C0∥µ∥(A) < ∞ holds where C0 = 4, so that both are equivalent. Considering coordinate-wise inequalities one sees that the same equiv- alence statement (with a constant Cn in place of C0 above) holds in n-dimensions, but in infinite dimensions the constant Cn → ∞, and so a different approach is needed. In fact it can be shown that ∥Z∥(S) < ∞ always, and one may have |Z|(S) = ∞. However the Vitali variation is σ-additive but the Fréchet variation is only σ-subadditive. Since X can be a Banach or a Fréchet space, a finer analysis is needed to define integrals in infinite dimensional X , for instance if X is any of the spaces Lp(P ), p ≥ 0 on a probability triple (Ω, Σ,P ). A classical observation due to I. Kluvánekimplies that a (vector) measure Z : A → X , a Banach space, where A is a σ-ring actually has its support in a set S0 ∈ A so that it is effectively defined on the (trace) σ-algebra A(S0) and so a vector measure into a Banach space defined on a σ-ring or algebra makes no difference and hence the words σ-ring and algebra may be used without distinction and this will be done below. Also it follows that a vector measure Z : S → X , a Banach space, defines a signed measure x∗(Z), and hence is bounded for each x∗ ∈ X ∗, the dual ∥ ∥ space, implying that (by the uniform boundedness theorem) sup∥x∗∥≤1 Z(A) < ∞,A ∈ S which shows that the Fréchet variation of a vector measure is always finite on a σ-ring or -algebra. It is seen that this (Frećhet)variation is σ-subadditive but generally not countably additive in infinite dimensional (Banach) spaces. If the range space of the vector measure is of the form B(X , Y), the vector space of bounded linear operators, which is a Banach space under the uniform (operator) norm the above definitions of Vitali and Fréchet variations apply, but using the special structure of the range space one may consider an analog of Fréchet’s variation (suggested by Dinculeanu) replacing the scalars of (4) by some vectors as follows. Observe that in Definition 1 if the multiplying scalars ai are replaced by the vectors aix, x ∈ X and the range space is Z = B(X , Y) so that ∥Z∥(Ω) is still ∗ finite, but if the range Y of Z, is the adjoint space X , then changing ai to xi, i = INTEGRATION WITH VECTOR VALUED MEASURES 5431

1, ··· , n alters the situation and the resulting Fr´echet and Vitali variations (ﬁnite or not) become equal, so that |Z|(Ω) = ||Z||(Ω). This is seen as follows. Since ∥Z∥(A) ≤ |Z(A)|,A ∈ Σ is always valid, for the opposite inequality consider for ε > 0 and integer n, with the Hahn-Banach theorem, an xi ∈ X , i = 1, ··· , n such ∈ S ∥ ∥ ≤ ⟨ ⟩ ε that for disjoint Ai (A), one has Z(Ai) xi,Z(Ai) + n , so that

∑n ∑n ∥Z(Ai)∥ ≤ ⟨xi,Z(Ai)⟩ + ε ≤ ∥Z∥(A) + ε. (5) i=1 i=1

This implies, on taking the supremum suitably, that ∥µ∥(·) = |µ|(·) as asserted. It will now be shown that this strengthening excludes one of the most important applications of the subject, namely the Brownian motion (BM). This (known) result will be sketched here for a clarification of the ensuing discussion. 2 Let {Xt, 0 ≤ t ≤ 1} be a BM-process with variance parameter σ = 1. Let Yin = Xi/2n −X(i−1)/2n be the increments which are independent Gaussian random variables with means zero and variances 2−n. Then a simple computation shows that ∑ n ∑ n 2 2 −n+1 | 2 2 − | ≥ V ar( i=1 Yin) = 2 . If An,ε = [ i=1 Yin 1 ε], then the Gaussian property n 2 of the Xt easily implies that P (An,ε) ≤ 2/2 ε using the Cebyˇshevinequality.ˇ Hence by the Borel-Cantelli lemma P (An,ε, i.o) = 0 where i.o. stands for infinitely often which is ‘limsup’ of sets. Thus the complementary event satisfies

∑2n | 2 − | ≥ P [ Yin 1 < ε, n n0] = 1. (6) i=1

But this means that, on using another key property of the BM namely the almost sure uniform continuity of its paths on each interval, one has the following inequality with probability one:

∑2n 2 1 = lim [Xi2−n − X(i−1)2−n ] n→∞ i=1

≤ lim max |Xi2−n − X(i−1)2−n | n→∞ 1≤i≤2n ∑2n × |Xi2−n − X(i−1)2−n |. i=1

Note that the first factor tends a.e. to zero (the uniform continuity of the BM process) and consequently the second factor must tend to infinity, with probability n n one. This implies that the set function Z defined by Z([(i−1)/2 , i/2 ]) = Xi2−n − X(i−1)2−n , which is seen extendable to a vector measure on the Borel σ-algebra of [0, 1] denoted again by Z cannot have a finite (Vitali) variation. But note that this Z actually takes values in a Hilbert space and the thus defined (vector) measure has (as Banach valued) a finite Fréchet variation. (It can be shown that the Vitali variation of BM on every subset of [0, 1] of positive Lebesgue measure is infinite!) In order to include this important process, it is necessary to proceed with the classical Fréchet variation and produce a ‘control’ measure that governs Z. The following account may be considered as a complement to a detailed analysis on random measures presented in (Rao (2009)) and many applications in (Rao 5432 M. M. RAO

(2012)). Since the general point of view and its details can be found there, some of the arguments will be shortened. 2. The vector valued integral. The following basic result was established by Bartle, Dunford, and Schwartz and it plays a key role in a large part of this work. An alternative argument based on the Kakutani representation of an abstract (L1)- space will be sketched for a comparison. Theorem 2. If ν : S → X is a vector measure where X is a B-space and S is a σ-(ring or) algebra, then there exists a finite measure λ : S → R+ which dominates ν. Moreover, (a) λ(A) ≤ ∥ν∥(A), and (b) limλ(A)→0 ∥ν∥(A) = 0.[λ is often termed a controlling measure of ν.] Outline of proof. It is enough to consider the result for real B-spaces, since the result can then be extended to the complex case as in the Hahn-Banach theorem. Thus let B(T, T ) denote the space of real σ-additive measures on (T, T ) which is a vector lattice under the partial order ≺ where for µi, i = 1, 2, µ1 ≺ µ2 if µ1(A) ≤ µ2(A),A ∈ T and the norm being the total variation. Here max(µ1, µ2), and min(µ1, µ2) are used in the partial ordering. Since the variation norm is additive on positive elements and if µ1, µ2 have disjoint supports, then these satisfy ∥µ1 + µ2∥ = ∥µ1 − µ2∥. Thus B(T, T ) is an abstract (L)-space with a unit which is any measure having support Ω. Now by Kakutani’s (classical) concrete representation, 1 the set B(T, T ) is isometrically isomorphic to an L (S1, S1, λ) on a Stone space 1 S1. If τ : B(T, T ) → L (S1S1, λ), is the isometric onto∫ mapping given by this ∈ B T ∈ S representation, then for each µ (T, ) let (τµ)(A) = A fµdλ, A 1. It defines 1 a unique fµ ∈ L (λ) and sets up an isomorphism. Now consider the set of scalar measures {x∗(ν), x∗ ∈ X ∗} as elements of B(T, T ). −1 ∗ If µ0 = λ◦τ , then by the above representation all the scalar measures x (ν) vanish on µ0-null sets. Consider the Carathéodory generated measure λ0 using µ0 which then dominates all x∗(ν), x∗ ∈ X ∗. This implies by the Orlicz-Pettis theorem that the correspondence is uniform and then this λ0 qualifies to be a controlling measure of ν.

Remark. There are several other proofs of finding the ‘controlling measure’ λ0, but all the methods depend on deep results of abstract measure theory (just as the Kakutani representation here). Also none is constructible and all assert only the existence. The∫ availability of such a measure is needed to get the well-definedness of the integral A fdν for all bounded (and then other) measurable scalar functions f on S. Now with the availability of such a controlling (positive finite) measure, one can obtain the key limit theorems for such integrals of scalar (measurable) functions, using the Vitali-Hahn-Saks-Nikodýmtheorem. The following result, from Dunford and Schwartz, is useful for the stochastic integration theory. Here the scalar measurable function f : S → R(C) is defined if there is a sequence ∫fn, n ≥ 1 of simple S → { ≥ } ( )-measurable functions such that fn f pointwise,∫ and A fndν, n 1 is X ∈ S Cauchy in the norm of . This limit is, by definition, A fdν, A for a vector measure ν : S → X . The existence of the integral is thus established using the availability of a controlling measure λ :Σ → R+. The necessary results for the above defined vector integral are of interest in stochastic and other applications as in: Theorem 3. The following properties of the above defined (vector or ν : S → X ) INTEGRATION WITH VECTOR VALUED MEASURES 5433 integral hold: ∫ 7→ ∈ X ∈ 1 (a) The mapping νf : A A fdν is linear as a mapping of f L (ν), the vector space of ν-integrable scalar functions, and defines a vector measure in A ∈ S. 1 (b) If fn → f,a.e. [ν] (i.e., in [λ], the controlling measure), where fn, f∫ ∈ L (ν), ≥ and the fn, n 1 are uniformly integrable in the∫ sense that lim∫ ∥ν∥(A)→0 A fndν = ∈ 1 0 uniformly in n, then f L (ν) and limn→∞ A fndν = A fdν. [A sufficient condition for this is that there is a ν-integrable g such that |fn| ≤ |g|, a.e. (λ0), the control.] (c) If Y is another Banach space and T : X → Y is continuous linear and ν is as above, with f ∈ L1(ν) , then ν˜ = T (ν):Σ → Y is a vector measure with∫ the same∫ control measure so that f is ν˜ integrable and moreover it holds that T ( A fdν) = A f dν˜. A proof of this result which summerizes the essential property of the vector integral that will be used below, is based on the work given in Section IV.10 of Dunford-Schwartz(1958) volume. It is stated in a form immediately useful for stochastic situations. From a well known result of 1940, due to M. M. Day, it is seen that the spaces Lp([0, 1]), 0 < p < 1 have no nontrivial (i.e., other than 0) continuous linear functionals, so that the above argument of Theorem 2 is not applicable and hence the existence of controlling measures cannot be deduced from it. But these spaces are important for stochastic integrals where the random measure ν is Lp-valued, with p ≥ 0. When once such an integral is at hand, it can be greatly extended with Bochner’s boundedness principle (Bochner (1956, Sec. 6.2) and Rao (2004)). Recalling specializations of the measure space (S, S, µ) with S = [0, 2π] and µ as normalized Lebesgue measure, one can show that the Hardy spaces Hp, 0 < p < ∞ as completions of trigonometric (or those of eiθ, 0 ≤ θ ≤ 2π) polynomials in the metric of the Lp-spaces, so that one has not only that Hp, p ≥ 1, to be Banach spaces, but for 0 < p < 1 there exist sufficiently many continuous linear functionals that separate points of (0, 2π] and consequently one may expect controlling measures for those valued in these Hp-spaces. However this is still not sufficient to invoke the Hahn-Banach theorem here to extend Theorem 2, since there exist subspaces M ⊂ Hp, 0 < p < 1 such that for no x∗ ∈ ((Hp)∗ − {0}) one has x∗(M) = {0}, (cf. Duren, 1970, p. 119). So another approach of getting a control measure for measures valued in Lp(µ), p ≥ 0 is desired. An analysis of the latter spaces was given in Talagrand (1981), also independently in Kalton-Peck-Roberts (1982). A more inclusive class of (nonlocally convex) Fréchet spaces will now be considered and the corresponding vector and stochastic integrals defined for them. The difference here is not changing the measure space but to analyze the structure of a large class of F-spaces and introduce another equivalent metric (actually a norm) and use their isomorphic properties to overcome the obstruction found in the Hp, 0 < p < 1 context. This is done in the next section. 3. Nonlocally convex Fréchet spaces with the Hahn-Banach property. As noted before, any linear metric space {X , d(·, ·)} can be assumed to have an equivalent translation invariant metric d(· − ·) by the theorem of Kakutani (1936) which was extended to hold also when completed, hereafter termed Fréchet spacs, by Klee (1952). So such metric spaces can and will be taken to have (translation) invarient metrics without comment. A class of these spaces that are not necessarily locally convex, will be constructed. These are function spaces for (stochastic) 5434 M. M. RAO integration. They include Lp(µ), 0 ≤ p < 1. For this purpose, consider a concave φ-function φ : R+ → R+, φ(0) = 0,which is nondecreasing, nontrivial, continuous, and extended to R by setting φ(−x) = φ(x). On a measure space (S, S, µ) φ consider the vector∫ space L (µ) = {f : S → R, ρφ(αf) < ∞} for some α = αf S where ρφ(f) = S φ(f)dµ for -measurable f. Next define an invariant metric, (also termed an F -norm) as follows:

|f| ∥ · ∥ : f 7→ ∥f∥ = inf{β > 0 : ρ ( ) ≤ β} (7) φ φ β which is seen to be an invariant metric but need not be a norm. On the other hand (7) and the functional

|f| N : f 7→ N (f) = inf{k > 0 : ρ ( ) ≤ 1} (8) φ φ φ k are norms provided φ is convex. It is then verified (nontrivially) that both {Lφ(µ), ∥· φ ∥} and {L (µ),Nφ(·)} are complete metric (or Fréchet) and Banach spaces respectively when equivalent functions are identified. If φ is convex, the functionals ∥ · ∥φ φ and Nφ(·) are equivalent in the sense that a sequence {fn, n ≥ 1} ⊂ L (µ) is con- vergent in one metric if and only if the same holds in the other. These are not obvious and the details can be found, e.g., in the book by Rao and Ren (1991, p. 400). Now φ is said to satisfy the ∆2-condition if φ(2x) ≤ Kφ(x), x ≥ 0 for some fixed constant K > 0, and this includes Lp(µ), p ≥ 0 where for p = 0 one x ≥ takes φ0(x) = 1+x , x 0, so that the ∆2-condition is satisfied in both cases and Lφ0 (µ) = L0(µ) since the metric topology of the former is equal to that of the latter with convergence in measure as its metric, φ0 is also concave increasing. For each f : S → R, consider its decreasing rearrangement, namely if λf (y) = ∗ µ({s : |f(s)| > y}), y ≥ 0. Set f (t) = inf{y : λf (y) ≤ t}, t ≥ 0 with inf{∅} = ∞ as usual, and define ∫ ∞ ∥ · ∥′ 7→ φ : f φ(λf (y))dy (9) 0 for any measurable f. Then one has the following nontrivial fact: ∥·∥′ { φ ∥· Proposition 4. The functional φ is a norm. If also µ is σ-finite, then L (µ), ∥′ } φ is a Banach space. ′ This depends on the property that φ is concave so its (left) derivative∑ φ is de- n creasing,∑ and the triangle inequality is first verified for functions f = i=1 iχAi ; g = m j=1 jχBj and using the monotonicity properties of the norm and some further work. It is shown that the triangle inequality and completeness of Lφ(µ) are valid. The following important result, due to Steigerwalt and White (1971) is needed for the main result of this section. Hereafter let µ be σ-finite, and for applications here it will be finite. { φ ∥ · ∥′ } Theorem 5. If φ is a concave increasing φ-function as above,then L (µ), φ is a Banach space. Moreover, if φ(x) ↗ ∞ as x ↗ ∞, then its adjoint space is φ ∗ ∼ representable as (L (µ)) = Aφ(µ) where the latter is a space of µ-continuous scalar set functions ζ(·) such that

|ζ(A)| ∥ζ∥(S) = sup{ : 0 < µ(A) < ∞,A ∈ S}. (10) φ(µ(A)) INTEGRATION WITH VECTOR VALUED MEASURES 5435

↗ ∞ ↗ ∞ φ(u) · Also if φ(t) as t , but limu→∞ u = 0, then ζ( ) is R-N diﬀerentiable dζ for µ and one has, with g = dµ , ∫ x∗(f) = fgdµ, f ∈ Lφ(µ) (11) S

∥ ∗∥ ∥ ∥′′ ∥ ∥ uniquely, and x = g φ = ζ (S) given by (10). This key result is used to obtain the following important vector integral defined for measures valued in Lφ(P ), based on a probability space (Ω, Σ,P ) in extending Theorem 2. It includes certain non-locally convex Fréchet function classes such as Lp(P ), 0 ≤ p < 1 on a probability space (Ω, Σ,P ), as ranges of random measures Z(·) regarded as vector measures on (S, S) with values in Lφ(P ) on a probability space (Ω, Σ,P ). Theorem 6. Let (S, S) be a measurable space and Z : S → Lφ(P ) be a random measure into the above type metric space on a probability triple (Ω, Σ,P ) where φ is an increasing φ-function which is either concave or convex. Then∫ there is a finite 7→ measure µ, controlling Z, and the vector integral T : f S f(s)dZ(s) is well- defined for all f ∈ Lφ(µ)(or ∩ L1(µ)). In particular it is defined for any bounded measurable f ∈ L1(µ). Remark. If φ is convex, then L1(P ) ⊃ Lφ(P ) for any probability P , but if φ is φ ⊃ 1 |x| ⇒ φ concave increasing, then L (P ) L (P ). Recall that φ(x) = 1+|x| L (P ) = L0(P ), the space of all random variables. Proof. Since it is known that Lφ(P ) is a Banach space when φ is a convex φ- function, so the theorem is a consequence of Theorem 2, it suffices to consider the case that φ is concave increasing. But now the above noted theorem of Steigerwalt ∥·∥′ φ and White (1971) implies that under the equivalent metric φ (norm now) L (P ) becomes a Banach space and hence by Theorem 2 again, there exists a finite positive measure µ0 that controls Z. Then the Z-integrable functions include all the bounded elements of Lφ(µ). Also in Stiegerwalt and White (1971, p.142), it∫ was observed that simple functions are dense in this space, and then the integrals S fdZ are defined for all f ∈ Lφ(µ). Only the existence of a (finite) control measure is needed. In particular all the bounded functions of L0(µ) have well-defined Z-integrals, and consider the space∫Lp(Z) of Z-integrable functions for members of Lp(µ). Then the 7→ ∈ φ ∩ 1 mapping T : f S f(s)dZ(s) is defined for all f L (µ) L (µ) and it is linear. Further details can be omitted. Discussion. 1. In defining the vector integral only the existence of a controlling (positive and finite) measure is needed and used. It is also possible to define an Lp(Z) and analyze their structure as was considered in Panchapagesan (2008) for p ≥ 1. This and other possibilities of the vector integration could be studied in this context. 2. A class of φ-functions that include t 7→ |t|p, 0 < p < 1 and many others is given by φ(x) = Φ−1(x), x > 0 where the Φ(·) are symmetric, convex, vanishing only at Φ(x) ∞ Φ(x) zero, and satisfying limx→∞ x = and limx→0 x = 0, called (continuous) −1 N-functions. The resulting spaces LΦ (µ) on a finite measure space (S, S, µ) were detailed in Rao and Ren ((2002), Sec. 10.5) and they constitute a very large concrete class of those in the theorem, and useful for applications. 5436 M. M. RAO

3. It was detailed in the above reference that this special class of Lφ(µ)- spaces admits an equivalent simpler F -norm (or invariant metric) given by: ∫ −1 ∥f∥Φ−1 = Φ (|f|)dµ, (12) S for which Theorem 5 applies. So there exists a norm in the space relative to which −1 this space becomes a B-space, and hence the∫ measure Z : S → LΦ (P ) has a 7→ ∈ φ ∩ 1 controlling measure. Thus the integral f S f(s)dZ, f L (µ) L (µ) defines elements of L1(P ) on a probability space. This is a useful and significant concrete class of scalar functions for which integration relative to vector or random measures can be considered. It should be observed, however, that the class of φ-functions is much larger than that given by φ = Φ−1 since the class properly includes both the p convex and concave cases. For instance, φ(x) = e|x| − 1, 0 < p < 1 is a φ-function which is just positive increasing, of the admitted class but is neither convex nor concave. 4. It may be observed that the concentration of spaces considered are linear metric spaces in which it is assumed that these are translation invariant. This is not a restriction. As already noted it was shown by Kakutani (1936) that a linear metric group always has an equivalent invariant metric, and that result was extended by Klee (1952) to a completion of the linear metric space in question so that the restriction to complete invariant linear metrics (also termed Fréchet spaces) is actually a general class. 4. Some Fréchet space integrals regardless of R-N properties. Com- plementing the preceding section, let φ : R → R+ be a symmetric φ-function, φ(0) = 0, and Lφ(P ) be the corresponding F -space on a probability space (Ω, Σ,P ). If X,Y ∈ Lφ(P ), then by linearity X + Y ∈ Lφ(P ), and obviously this does not always imply either of X,Y need be in the space. But when they are mutually independent the situation suddenly changes. In fact consider the set An = [|Y | < n] and observe that it is independent of X or a Borel function of it. Hence using the fact that X + Y ∈ Lφ(P ) so that E(φ(a(X + Y ))) < ∞ for some scalar a which for convenience can be taken as unity, one has

∞ > E(φ(X + Y )) ≥ E(φ(|X| − |Y |)) ≥ | − | E(φ( X n )χAn )

= E(φ(X − n))P (An), (13)

φ by independence or that X − n ∈ L (P ) for an n > 1 so that P (An) > 0, whence X ∈ Lφ(P ). Similarly Y is also in the space. This fact will be used below in conjunction with a related infinitely divisible property of the vector measure Z(·). First note that Lφ(P ) ⊂ L0(P ) and hence Z(·) is also valued in the latter space which is identifiable as Lφ0 (P ) with the continuous concave increasing φ-function 7→ |x| φ : x 1+|x| and by Theorem 5 it is isomorphic to a Banach space. This implies by Theorem 6 that there is a finite controlling measure λ for Z(·): S → Lφ(P ), even without using independent valued condition. But with the latter, it is possible to find a more concrete controlling measure allowing a detailed analysis. As seen in the Remark following Theorem 6, the largest vector space Lφ0 (P ) = 0 L (P ), the space of (finite) random variables, is an F-space where φ0 is still concave INTEGRATION WITH VECTOR VALUED MEASURES 5437 increasing. By the first part of Theorem 5, this space as already noted is isomorphic to a Banach space although itself is only an F-space. Hence by Proposition 4 and Theorem 2, one has for a vector measure Z : S → Lφ0 (P ), treated as σ-additive in the topology of the latter, there∫ exists a controlling measure λ so that the Dunford- ∈ S Schwartz (or random) integral A fdZ, A , is defined and is linear on all simple and then bounded S-measurable functions. It is desirable to characterize this set at least for some classes of random measures. If Z(·) has independent values and (S, S, µ) is a (topological) Borel space where points are measurable, then for each A ∈ S, one can regard Z(A) as the (finite or countable) sum of independent random variables in Lφ(P ), a subspace of Lφ0 (P ), and the range space may be characterized if Z has independent values for disjoint sets, using the F-norm of the range space. This useful identification will be described extending some of Urbanik’s(1967) early analysis. Take S ⊂ R for simplicity hereafter. ∫ ∫ The independent value property of Z is easily seen to imply that A fdZ and B gdZ are independent random variables when f and g have µ-disjoint supports. Since Z(A) need not have any moments, its properties can often be analyzed through its Fourier transform (FT). The independent values of Z, its σ-additivity, and the fact that S has points measurable, indicates that Z(A) is infinitely divisible in the ‘generalized sense’ and by a known theorem due to J. L. Doob such Z(A) is infinitely divisible in Lévy’ssense. Thus without further discussion, assume that Z(A),A ∈ S is∫ an infinitely divisible random field. Then the FT of Z(A), denoted φZ(A)(t) = itZ(A) Ω e dP , has the following well-known Lévy-Khintchine representation: ∫ { itu − − itu φZ(A)(t) = exp itµ(A) + (e 1 2 ) S 1 + u (1 + u2) × dG (u)},A ∈ S, (14) u2 Z(A) where A is a bounded Borel set and GX (·) ≥ 0, is a bounded increasing function and µ ≥ 0, a measure, so that the integral is well-defined. If µ0(A) = |µ|(A) + GA(∞) then Z(·) is dominated by the measure µ0 which may be taken finite, and it controls Z(·) so that the vector integrals of Z for scalar functions that are µ0-integrable are defined. This is a ‘concrete’ candidate controlling the (vector or) random measure Z. Now a characterization of the set of Z- integrable functions is presented for a class of infinitely divisible Z(·) which explains the structure of the problem of vector measure integration. It may be noted that a vector measure in a Banach space is bounded, as a consequence of the uniform boundedness principle that is based on the Hahn-Banach theorem. This is still true if the measures are valued in L0(P ) spaces, by a result proved independently in Talagrand (1981) and Kalton-Peck-Roberts (1982), included in Proposition 4 above. It was already used in Garling (1975, p.36) in order to invoke the lifting theorem on Z for his work. This actually follows from the earlier results of Steigerwalt and White (1971) also. Thus the vector integral can be defined for all Lφ(P )-valued measures Z(·) since the latter spaces are contained in L0(P ) with the Fréchet norm in the form of convergence in probability. But Z with independent values has a simpler control measure, due to the infinite divisibility property. It will now be specialized to stable classes contained in the infinitely divisible family, and to show that the range space is actually a generalized Orlicz space. This analysis leads to an interesting separate study, and it will be sketched here. The work will be summerized as the main result later. 5438 M. M. RAO

Recall that P. L´evyin 1924 already introduced the class of stable random variables X, predating the inﬁnitely divisible class, which may be described in the current terminology, through their characteristic functions or FT’s having the following representation:

α φX (t) = exp[λ(itγ − |t| + itΓ(t, α, β)], (15) where { (|t|α−1 − 1)β tan( πα ), if α ≠ 1 Γ = 2 − 2 | | β( π ) log t , if α = 1. A simple example of this is the class of Pareto distributions of interest in Finance and Mathematical Economics whose density in the symmetric case is given by: α f (x) = |x|−(α+1); |x| ≥ 1, α1 2 and vanishes elsewhere. In the general case of (15) the set {λ, α, β, γ} of parameters of the family must satisfy |β| ≤ 1, 0 < α ≤ 2, λ, γ ∈ R, but the FT is not necessarily continuous in the parameters (see Zolotarev (1056) for details). Moreover, if |α| < 1 and β = 1(or = −1) the random variable X (or its density F ′) is zero on the negative(respectively positive) axis. It suﬃces to consider here the positive case (i.e., β = 1) when X = Z(·), the random measure, and its integrals. (See also Zolotarev (1956), p.11 and p.80.) This aspect of positivity is incorporated in the derivation, of not its FT, but by considering the Laplace transform on R+, namely −Z(A) LZ(A)(t) = E(e ), t ≥ 0, which exists and its L´evy-Khintchine analog was given in Feller (1966; p.426). It has a special form of interest here. The Laplace transform −ψA(u) of Z(A) can be expressed as LZ(A)(u) = e where ∫ ∞ 1 − e−tx ψ (t) = β t + dµ(x),A ∈ S, t ≥ 0, (16) A A − −x 0 1 e

+ with β(·), µ(·) as finite measures on S and B(R ) respectively. Using an extension of this and a related result by Urbanik (1967) in the class of infinitely divisible distributions, an interesting modification was given by Garling (1975), who obtained the following useful form for ψZ(A). It is represented as:

∫ ∫ ∞

ψZ(A)(t) = βAt + ( k(t, x)g(y, dx))dµ(y), (17) A 0 where β(·), µ(·) are finite measures on S, and 0 ≤ g(x, ·) ≤ 1 is increasing for each x ∈ S, with 0 ≤ k(·, x) ↑ and k(t, ·) is measurable. Moreover if the nonnegative infinitely divisible random measure Z(·) is translation invarient when S is a semi- group so that Z(A + x) = Z(A),A ∈ S, x ∈ S, then g is just a measure on S. Thus (17) holds for all nonnegative infinitely divisible random measures Z(·) with values in L0(P ) on a probability space (Ω, Σ,P ). The set of functions integrable relative to such a random measure Z turns out to be a function space, called the Musielak-Orlicz space. It is not locally convex but is an F-space. In (17) it is seen that β(·) is a measure and this corresponds to a nonrandom additive component of Z(·) since the FT of the sum of independent random variables is the product of the FTs of the components, so that subtracting such a constant measure from Z(·) does INTEGRATION WITH VECTOR VALUED MEASURES 5439 not change the main problem, and the new measure will still be infinitely divisible as before, and the constant term in (17) will be absent. Assuming this is done, the new (‘centered’) random measure will be denoted by the same symbol, with β(·) = 0 in (17). This useful simplification, first noted by Garling (1975), will be in force hereafter, and the class of functions that are integrable for this (reduced) Z(·) will be considered here. ∫ · · If K(t, y) = R+ k(t, x)g(y, dx), then K(t, ) of (17) is increasing and K( , y) is measurable for S. Further K(t, 0) = 0,K(t, 1) = 1, and define K(t, −y) = ∫K(t, y) to extend the definition to all of R. If one defines the ‘modular’ ρ(f) = | | R → R LM S ρ(t, f(t) )dµ(t), for f : , measurable and considers the space (µ) = {f : ρ(af) < ∞, a ∈ R}, then it can be seen that LM (µ) is a linear space. If ∥ · ∥ 7→ { f ≤ } ∥ · ∥ M : f inf a > 0 : ρ( a ) a as in (7), then it can be verified that M M is a metric and the space {L (µ), ∥ · ∥M } is not only a vector space, but is also complete although not locally convex. The proof is analogous to the earlier case, but the argument in the current somewhat more general case is also similar. One can find complete details of this general case in Shiragin (2007; Chapter 8). Although he assumed that K(t, ·) is convex, it is not essential. The proof goes through as in Rao and Ren (1991, Theorem 10.1.2). This is often called the Musielak-Orlicz space. This relaxation was shown slightly differently in the original works of these authors as well. Garling’s proof assumes the usual “∆2 condition”, which slightly simplifies the work but now seen as not essential. The preceding analysis with simple adjustments can be summerized as:

Theorem 7. Let Z : S → L0(P ) be a nonnegative random measure that is infinitely divisible and has no constant component measure so that its Lévy-Khintchineform of the Laplace transform is given by (17) without the constant measure β(·). Then the class of Z(·)-integrable scalar functions constitute a Muslieak-Orlicz space on a finite measure space (S, S, µ) where S ⊂ R. It can be shown that if Z(·) is translation invariant then the integrable functions for Z constitute a generalized Orlicz space (i.e., K(t, ·) is independent of t) that is not necessarily locally convex (as in Proposition 4). A detailed analysis, if Z is restricted to be translation invarient, was considered by Urbanik (1967) where its application to prediction problems on processes that are strictly stationary in the sense of G. D. Birkhoff and A. Ya. Khintchine which have one moment.

5. Final remarks. It may be observed that although random measures may be considered as a subclass of vector measures, with values in topological vector spaces, the theory in non-Banach spaces encounters many diﬃculties. But the subclass of ‘stochastic or random measures’ brings in many specialized properties contributed by the underlying probabilistic structures. For instance, positive inﬁnitely decom- posable class, translation invariance, or stationarity and the like. They can also take values in function spaces that are themselves vector valued and this uses both the vector (metric) space analysis as well as probability properties. This points out some of the many possibile applications in this study. With translation invariance specialization for Z(·), Garling (1975) has studied order preserving maps and related embedding properties of considerable interest. The preceding work thus gives a glimpse of many such possibilities with random measures. Finally, I should acknowledge Prof. I. Shragin for kindly sending me his work [17]which was not available for me. 5440 M. M. RAO

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