# Riesz Representation Theorems For Positive Linear Operators

## Transcript Of Riesz Representation Theorems For Positive Linear Operators

arXiv:2104.12153v4 [math.FA] 5 Jan 2022

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

MARCEL DE JEU

Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands and

Department of Mathematics and Applied Mathematics, University of Pretoria, Corner of Lynnwood Road and Roper Street, Hatﬁeld 0083, Pretoria, South Africa

XINGNI JIANG

College of Mathematics, Sichuan University, No. 24, South Section, First Ring Road, Chengdu, P.R. China

ABSTRACT. We generalise the Riesz representation theorems for positive linear functionals on Cc(X ) and C0(X ), where X is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space E. The representing measures are deﬁned on the Borel σ-algebra of X and take their values in the extended positive cone of E. The corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of X . Results are included where the space E need not be a vector lattice, nor a normed space. Representing measures exist, for example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras.

1. INTRODUCTION AND OVERVIEW

Let X be a locally compact Hausdorff space, and let π : Cc(X ) → Ê be a posit-

ive linear functional. The Riesz representation theorem asserts that there is a unique regular Borel measure on the Borel σ-algebra of X , such that

(1.1)

π( f ) = f dµ

X

E-mail addresses: [email protected], [email protected]

2010 Mathematics Subject Classiﬁcation. Primary 47B65; Secondary 28B15. Key words and phrases. Riesz representation theorem, measure, order integral, ordered vector space.

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partially

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RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

for all f in Cc(X ). In this paper, we establish analogous representation theorems for positive

linear operators π : Cc(X ) → E and π : C0(X ) → E, where E is a (suitable) partially ordered vector space, while giving explicit formulas for the measures of open and of compact subsets of X .1 Results are included where the space E need not be a vector lattice, nor a normed space. As will become clear, representing measures exist for, e.g., positive linear operators into Banach lattices with order

Ê continuous norms, into the regular operators on KB-spaces, into the self-adjoint

operators on complex Hilbert spaces, and into JBW-algebras. When E = , the results specialise to the classical Riesz representation theorems.

In the general setting of the present paper, the measure µ in equation (1.1) takes its values in the (extended) positive cone of E, and the integral in question is an order integral. Such measures and their order integrals are the subject of [10], which extends earlier work by Wright. The results for the order integral in [10] are fairly complete and also include convergence theorems that can be of use in applications, with the existence theorems in the present paper as a starting point.

A possible multiplicativity of the linear operator π is not an issue in the current paper: being positive and linear is enough. In the sequel [12], we shall consider positive algebra homomorphisms from Cc(X ) or C0(X ) into partially ordered algebras. The representing measures from the current paper can then be shown to be spectral measures that take values in the algebras. It will be seen in [12] that the ensuing existence theorems for abstract spectral measures immediately imply the classical ones for representations of (the complexiﬁcation of) C0(X ) on complex Hilbert spaces, and for positive representations of C0(X ) on KB-spaces in [13]. The up-down theorems that are established in [12] for general partially ordered algebras yield the familiar results for Hilbert spaces as special cases. In [11], which is another sequel to the present paper, we shall be concerned with representation theorems for vector lattices (resp. Banach lattices) of regular operators from Cc(X ) and C0(X ) into Dedekind complete vector lattices (resp. Banach lattices with order continuous norms) in the spirit of [4, Theorem 38.7].

We shall discuss the relation between the present paper and existing representation theorems for positive linear operators in the literature at the end of this introduction. There appears to be no previous work in the vein of the sequels [11] or [12] to the present paper.

This paper is organised as follows. Section 2 contains the necessary prerequisites from [10], including those on

measures and the order integral, and can serve as a summary thereof.

1In the course of the present paper, its prequel [10], and its sequels [11, 12], we shall encounter maps with Cc(X ) or C0(X ) as domains that are sometimes positive linear operators, sometimes vector lattice homomorphisms, and sometimes positive algebra homomorphisms. For each of these contexts, a canonical symbol for such maps could be chosen. However, since our results for these contexts are related, we have chosen to use the same symbol π throughout, thus keeping the notation as uniform as possible.

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

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In the preparatory Section 3, we introduce various types of regularity of measures on the Borel σ-algebra of a locally compact Hausdorff space that take their values in the extended positive cones of partially ordered vector spaces. A few auxiliary results for the remainder of the paper are also established,

Section 4 contains the proof of a representation theorem (see Theorem 4.2) for positive linear operators π : Cc(X ) → E, where E is (for all practical purposes) a Banach lattice with an order continuous norm. It is one of the essential ingredients for the sequel [11]. The possibly inﬁnite representing measure in it is always regular.

Since spaces of operators will only rarely satisfy the conditions on the space E in Section 4, other results are needed that do apply (at least) when E is a space of operators. Such results are to be found in Section 5, where representation theorems (see Theorems 5.4 and 5.6) are established when the codomain of the positive linear operator π : Cc(X ) → E is a monotone complete and normal space. The class of such spaces E is quite varied. We refer to Examples 2.5 and also to [10, Section 3] for examples, some of which were already mentioned above. When E consists of the regular operators on a Banach lattice with an order continuous norm, or of the self-adjoint operators in a strongly closed complex linear subspace of the bounded linear operators on a complex Hilbert space, then the representing measure in Theorem 5.4 has the familiar property of being σ-additive in the strong operator topology. The representing measures in Theorem 5.4 are ﬁnite (by assumption) and regular. Those in Theorem 5.6 can be inﬁnite, but regularity need then hold only locally. In Remark 5.8, we compare the applicability of the main representation theorems (Theorems 4.2, 5.4, and 5.6) for positive linear operators π : C0(X ) → E.

The ﬁnal Section 6 is concerned with positive linear operators π : C0(X ) → E. The domain C0(X ) is now a Banach lattice, and we can then exploit automatic continuity to derive representation theorems for such linear operators from those for their restrictions to Cc(X ). The representing measures thus obtained are all ﬁnite. Theorem 6.2 covers the case where E is a KB-space, and Theorem 6.8 applies to the larger class of quasi-perfect partially ordered vector spaces. In Theorems 6.10 and 6.12, the space E consists of the regular operators on a KB-space and of the self-adjoint operators in a strongly closed complex linear subspace of the bounded linear operators on a complex Hilbert space, respectively. In both cases, the representing measures are strongly σ-additive again. Furthermore, a number of SOT-closed subspaces of E that are naturally associated with π can be seen to coincide. As a consequence, the representing measure takes its values in the coinciding bicommutants of π(Cc(X )) and π(C0(X )).

We now give an overview of earlier work that we are aware of on Riesz representation theorems for positive linear operators from spaces of continuous functions into various types of partially ordered vector spaces. The space X is always a locally compact Hausdorff space, unless otherwise stated. As will become clear,

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RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

one can hardly speak of ‘the’ Riesz representation theorem for positive linear operators from Cc(X ) or C0(X ) into partially ordered vector spaces.

To start with, there is the seminal paper by Wright [23]. It has a representation theorem for a positive linear operator from C(X ), where X is compact, into a Stone algebra. A second paper [24] by the same author covers the case of a positive linear operator from C(X ), where X is compact, into a σ-Dedekind complete vector lattice. A third paper [25] contains a representation theorem for a positive linear operator from Cc(X ) into a Dedekind complete vector lattice. In a fourth [26], the existence is established of a representing Baire measure for a positive linear operator from C(X ), where X is compact, into a σ-monotone complete partially ordered vector space. When comparing the latter result to our Theorem 5.4, both have their strong points. Wright’s conditions on E are more lenient, but the space X has to be compact, which is essential to the proofs. Theorem 5.4, on the other hand, is valid for locally compact spaces, and also gives explicit formulas for the measure of open and of compact subsets.

In [16], one of Khurana’s papers in this direction, it is shown that a positive linear operator from Cc(X ) into a monotone complete partially ordered vector space can be extended to a σ-order continuous linear operator deﬁned on the bounded Borel functions with compact support; this goes in the direction of a Riesz representation theorem. In a second paper [17], a representation theorem is established, under certain additional conditions, for a positive linear operator from the bounded continuous functions on a completely regular Hausdorff space into a monotone complete partially ordered vector space. In [18], he proves a representation theorem for a positive linear operator from the continuous functions on a completely regular T1-space into a Stone algebra.

In [19], Lipecki shows the existence of a representing ﬁnitely additive measure for positive linear operators from the bounded continuous functions on an arbitrary topological space into a monotone complete partially ordered vector space; there is a second result for the bounded continuous functions when the topological space is normal.

In [8], Coquand gives a new proof of Wright’s result in [26]. The results cited above also include regularity properties of the representing measures, but not always the same properties. In cases where several results apply, it is, therefore, not clear whether the representing measures are necessarily equal. The results in the present paper have the advantage that the space X need not be compact and that they apply, amongst others, when E is a monotone complete normal space. This is a fairly large class of spaces, containing many spaces that are not vector lattices. Moreover, explicit formulas for the measures of open and of compact subsets of X are given. We are not aware of a similar combination of a reasonably wide range of applicability and concrete formulas in the literature on partially ordered vector spaces. The usefulness of this combination can be seen in, e.g, Theorems 6.10 and 6.12, which, though in two rather different contexts, are both virtually immediate consequences of one underlying general result. It will become even more pronounced in the sequel [12]

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

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when considering positive algebra homomorphisms from Cc(X ) and C0(X ) into monotone complete normal partially ordered algebras with monotone continu-

ous multiplications.

2. PRELIMINARIES

In this section, we collect some conventions, deﬁnition, notations, and preparatory results that will be used in the sequel.

The indicator function of a subset S of a set X is denoted by χS. We shall also write 0 for χ and 1 for χX . When X is a topological space, then we let Cc(X ), resp. C0(X ), denote the continuous real-valued functions on X that have compact support, resp. vanish at inﬁnity.

2.1. Partially ordered vector spaces. All vector spaces we shall consider are over the real numbers, unless otherwise indicated. An operator between two vector spaces and a functional are always supposed to be linear, but—when this notion is applicable—need not be bounded. We do not require that the positive cone E+ of a partially ordered vector space E be generating. Equivalently, we do not require that E be directed. We do require, however, that E+ be proper, i.e., that E+ ∩ (−E+) = {0}. All vector lattices are supposed to be Archimedean.

Deﬁnition 2.1. A partially ordered vector space E is called (1) σ-monotone complete if every increasing sequence {xn}∞ n=1 in E that is bounded from above has a supremum in E; (2) monotone complete if every increasing net {xλ}λ∈Λ in E that is bounded from above has a supremum in E; (3) σ-Dedekind complete if every non-empty at most countably inﬁnite subset S of E that is bounded from above has a supremum in E; (4) Dedekind complete if every non-empty subset S that is bounded from above has a supremum in E.

We shall employ the usual notation in which xλ ↓ means that {xλ}λ∈Λ is a decreasing net, and in which xλ ↓ x means that {xλ}λ∈Λ is a decreasing net with inﬁmum x. The notations xλ ↑ and xλ ↑ x are similarly deﬁned.

It was observed in [26, Lemma 1.1] that every σ-monotone complete partially ordered vector space E (and then also every monotone complete, σ-Dedekind complete, or Dedekind complete partially order vector space) is Archimedean, i.e., {ǫ x : ǫ > 0} = 0 for all x ∈ E+. We shall use this a number of times.

Vector spaces of operators between partially ordered vector spaces can inherit completeness properties from the codomains. In order to formulate this, we ﬁrst introduce some notation and terminology.

When E and F are vector spaces, then L(E, F) denotes the vector space of operators from E into F. An operator T ∈ L(E, F) between two partially ordered vector spaces is positive if T (E+) ⊆ F+, and regular if it is the difference of two positive operators. The regular operators from E into F form a vector space that is denoted by Lr(E, F). When E+ is directed, then every linear subspace

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RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

Ê of L(E, F) that contains Lr(E, F) is naturally partially ordered with the regular

positive operators from E into F, denoted by Lr(E, F)+, as its positive cone. We shall write L(E) for L(E, E), Lr(E) for Lr(E, E), and E for Lr(E, ). When E is a Banach lattice, then E coincides with the norm dual E∗ of E.

We can now state how completeness is hereditary. When E is a directed partially ordered vector space, and F is a partially ordered vector space that is monotone complete (resp. σ-monotone complete), then any linear subspace of L(E, F) containing Lr(E, F) is monotone complete (resp. σ-monotone complete); see [10, Proposition 3.1].

The monotone complete partially ordered vector spaces that are also normal will play an important part in this paper. We now proceed to deﬁne the latter notion.

Deﬁnition 2.2. Let E and F be partially ordered vector spaces, and let T : E → F be a positive operator. Then T is called order continuous (resp. σ-order continuous) if T xλ ↓ 0 in F whenever xλ ↓ 0 in E (resp. if T xn ↓ 0 in F whenever xn ↓ 0 in E). A general operator in Lr(E, F) is called order continuous (resp. σ-order continuous) if it is the difference of two positive order continuous operators. We let Loc(E, F ) (resp. Lσoc(E, F )) denote the order continuous (resp. σ-order

Ê continuous) operators from E into F; we shall write Eoc for Loc(E, ) and Eσoc

for Lσoc(E, ).2

It is easy to see that Loc(E, F ) and Lσoc(E, F ) are linear subspaces of Lr(E, F ). When E is directed, then they are partially ordered vector spaces with the positive order continuous operators (resp. the positive σ-order continuous operators) from E into F as positive cones, which are generating by deﬁnition.

Deﬁnition 2.3. Let E be a partially ordered vector space. Then E is called normal when, for x ∈ E, (x, x′) ≥ 0 for all x′ ∈ (Eoc)+ if and only if x ∈ E+. We say that E is σ-normal when, for x ∈ E, (x, x′) ≥ 0 for all x′ ∈ (Eσoc)+ if and only if x ∈ E+.3

The importance of normality for our work lies in the following result (see [10, Proposition 3.8]) that will be used quite a few times in the present paper.

Proposition 2.4. Let E be a normal partially ordered vector space. Suppose that {xλ}λ∈Λ is a net in E, and that x ∈ E.

(1) If xλ ↓, then xλ ↓ x if and only if (x , x ′) = infλ∈Λ(xλ, x ′) for all x ′ ∈ (Eoc)+.

2When E and F are vector lattices, where F is Dedekind complete, then the notions of order continuous and σ-order continuous operators in Deﬁnition 2.2 agree with the usual ones in the literature as on [27, p. 123]; see [10, Remark 3.5] for this.

3When E is a vector lattice, then the notion of normality in Deﬁnition 2.3 coincides with the usual one in the literature (see [1, p. 21], for example) that Eoc separates the points of E; see [10, Lemma 3.7] for this. It also follows from [10, Lemma 3.7] that a vector lattice E is σ-normal as in Deﬁnition 2.3 if and only if Eσoc separates the points of E.

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(2) If xλ ↑, then xλ ↑ x if and only if (x , x ′) = supλ∈Λ(xλ, x ′) for all x ′ ∈ (Eoc)+.

In the presence of monotone completeness, normality of a codomain is a hereditary property: when E is a directed partially ordered vector space, and F is a monotone complete and normal partially ordered vector space, then any linear subspace of L(E, F) containing Lr(E, F) is monotone complete and normal; see [10, Proposition 3.11].

Since both (σ-)monotone completeness and normality are properties that are inherited by partially ordered vector spaces of operators, it is easy to construct examples of monotone complete and normal spaces once one has such a space to begin with.

Examples 2.5. In [10, Section 3] we have included a number of examples of partially ordered vector spaces that are monotone complete and normal. These include (but are not limited to):

(1) Banach lattices with order continuous norms; (2) for partially ordered vector spaces E and F such that E is directed and

F is monotone complete and normal: every linear subspace of L(E, F) that contains Lr(E, F); (3) as a special case of part (2): the regular operators on a Banach lattice with an order continuous norm; (4) the real vector space that consists of the self-adjoint operators in a strongly closed complex linear subspace of the bounded operators on a complex Hilbert space; (5) JBW-algebras.4

2.2. Measures and order integrals. In this section, we shall brieﬂy summarise the relevant deﬁnitions and results for measures and order integrals from [10, Sections 4 and 6]. This extends earlier work by Wright and contains the usual Lebesgue integral as a special case. We refer to [10] for a discussion of the relation with Wright’s work.

We refrain from mentioning here in any detail the material on outer measures in [10, Section 5]. It is indispensable in the proof of Theorem 4.2, but it occurs only as an intermediate step and does not reappear.

Let E be a partially ordered vector space. As the case of the Lebesgue integral on the real line already shows, one cannot expect a representing measure for a positive operator π : Cc(X ) → E to be ﬁnite. This is why we extend E by introducing the set E := E ∪ {∞} as a disjoint union, and extend the partial ordering from E to E by declaring that x ≤ ∞ for all x ∈ E. Then E+ := E+ ∪ {∞} is the set of positive elements of E. The elements of E that are in E are called ﬁnite. One makes E into an abelian monoid by retaining the addition on E and deﬁning ∞ + x := ∞ and x + ∞ := ∞ for all x ∈ E. Then E+ is a

4We shall use [2,3] references for JBW-algebras. In these books, a JBW-algebra is supposed to have an identity element; see [3, Deﬁnitions 1.5 and 2.2]. In other sources, this is not supposed. However, as [15, Lemma 4.1.7] shows, the existence of an identity element is, in fact, automatic.

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RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

sub-monoid of E. When x, y ∈ E are such that x ≤ y, then x + z ≤ y + z for all

Ê Ê z ∈ E. We keep the action of on E, and deﬁne r ·∞ := ∞ for all r ∈ + \{0} Ê and 0 · ∞ := 0. Then r x ≤ r y for all r, s ∈ + and x, y ∈ E such that r ≤ s and

x ≤ y. Furthermore, r(x + y) = r x + r y, (r + s)x = r x + sx, and (rs)x = r(sx)

Ê for all r, s ∈ + and x, y ∈ E+. Such relations will be used in the sequel without

further reference. When being used to working with the extended real numbers, which are

still linearly ordered, it may be relatively easy to make mistakes when arguing

with the ordering of and the operations on E and E. It is for this reason that a fair number of technical tools have been collected in [10, Lemmas 2.3 to 2.5] that will be used repeatedly in the present paper and its sequels. Whenever

necessary, we shall be careful to indicate whether we are working in E or in E when speaking of order bounds, suprema, or inﬁma.

Now that we have the extended space E available, it is possible to deﬁne the measures that concern us. It was Wright who ﬁrst observed in [23] that equation (2.1), below, is the proper way to generalise the notion of σ-additivity from the real numbers to more general partially ordered vector spaces.

A measurable space is a pair (X , Ω), where X is a set and Ω is an algebra of subsets of X ; i.e., Ω is a non-empty collection of subsets of X that is closed under taking complements and under taking ﬁnite unions. For the moment, we can still work with algebras of subsets, rather than σ-algebras.

Deﬁnition 2.6. Let (X , Ω) be a measurable space, and let E be a σ-monotone

complete partially ordered vector space. A positive E-valued measure is a map

µ : Ω → E+ such that:

(1) µ( ) = 0;

(2) (σ-additivity) whenever {∆n}∞ n=1 is a pairwise disjoint sequence in Ω

with

∞ n=1

∆n

∈

Ω,

then

∞

∞N

(2.1)

µ

∆n =

µ(∆n)

n=1

N =1 n=1

in E.

Since µ is E+-valued, it follows from the σ-monotone completeness of E that

the supremum in the right hand side of equation (2.1) exists in E; see part (1) of [10, Lemma 2.5]. When µ(X ) ∈ E+, then we say that µ is ﬁnite, or that it

Ê is E-valued; when µ(X ) = ∞, then µ is said to be inﬁnite. As [10, Section 4]

shows, a good number of the properties of positive -valued measures hold in the general case as well, including even the Borel–Cantelli lemmas (see [10, Lemma 4.7]).

Next, we introduce the integral with respect to a measure. From now on, we suppose that Ω is a σ-algebra.

Ê A measurable function ϕ : X → + is an elementary function if it takes only

ﬁnitely many (ﬁnite) values. It can be written as a ﬁnite sum ϕ = ni=1 riχ∆i

Ê for some r1, . . . , rn ∈ + and ∆1, . . . , ∆n ∈ Ω. Here the ri are all ﬁnite, but it

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

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is allowed that µ(∆i) = ∞ for some of the ∆i. We deﬁne its (order) integral, which is an element of E+, by setting

o

n

ϕ dµ := riµ(∆i).

X

i=1

This deﬁnition is independent of the above expression for ϕ as a ﬁnite sum.

Ê When f : X → + is measurable, we choose a sequence {ϕn}∞ n=1 of elementary Ê functions such that ϕn ↑ f pointwise in +. We deﬁne the order integral of f ,

which is an element of E+, by

o

∞

f dµ :=

X

n=1

o

ϕn dµ.

X

Since Xoϕn dµ ↑, the σ-monotone completeness of E guarantees that this supremum exists in E+. It is independent of the choice of the sequence {ϕn}∞ n=1.

Ê We let L 1(X , Ω, µ; ) denote the set of all (ﬁnite-valued) measurable funcÊ tions f : X → such that o| f | dµ is ﬁnite. By splitting a function into its pos-

X

Ê itive and negative parts, the order integral is then deﬁned on L 1(X , Ω, µ; ). Ê We let B(X , Ω; ) denote the bounded measurable functions on X . For a ﬁnite Ê Ê measure µ, B(X , Ω; ) ⊆ L 1(X , Ω, µ; ).

In [10, Section 6.2], the monotone convergence theorem for the order integ-

ral is established. When E is σ-Dedekind complete, then Fatou’s lemma and the

dominated convergence theorem are also valid.

Ê The space L 1(X , Ω, µ; ) is a σ-Dedekind complete vector lattice, and the Ê order integral is a σ-order continuous positive operator from L 1(X , Ω, µ; ) Ê Ê into E; see [10, Proposition 6.14]. According to [10, Theorem 6.17], the space

L1(X , Ω, µ; ), where elements of L 1(X , Ω, µ; ) have been identiﬁed when

they agree µ-almost everywhere, is likewise a σ-Dedekind complete vector lat-

tice, and the order integral induces a strictly positive σ-order continuous op-

Ê erator Iµ from L1(X , Ω, µ; ) into E. When E is monotone complete and has Ê the countable sup property, then L1(X , Ω, µ; ) is a Dedekind complete vector

lattice with the countable sup property, and Iµ is order continuous.5

Remark 2.7.

(1) When E is a partially ordered Banach space with a closed positive cone, then every positive vector measure is a measure in the sense of Deﬁnition 2.6, but not conversely. Even when a measure falls into both categories, the domain of the order integral can properly contain that of

5As in [10, Section 6], we say that a partially ordered vector space E has the countable sup

property when, for every net {xλ}λ∈Λ ⊆ E+ and x ∈ E+ such that xλ ↑ x, there exists an at most countably inﬁnite set of indices {λn : n ≥ 1} such that x = supn≥1 xλn . In this case, there also always exist indices λ1 ≤ λ2 ≤ · · · such that xλn ↑ x. For vector lattices, this deﬁnition coincides with the usual one as can be found on, e.g., [5, p. 103].

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any reasonably deﬁned integral with respect to the vector measure using Banach space methods. We refer to [10, Section 7] for this discussion. (2) When E consists of the regular operators on a Banach lattice with an order continuous norm, or when E consists of the self-adjoint operators in a strongly closed complex linear subspace of the bounded operator on a complex Hilbert space, then the ﬁnite measures in the sense of Deﬁnition 2.6 are precisely the set maps µ : Ω → E+ with µ( ) = 0 that are σ-additive with respect to the strong operator topology on E; see [10, Lemmas 4.2 and 4.3].

3. MEASURES ON LOCALLY COMPACT HAUSDORFF SPACES

Ê There may be several measures on the Borel σ-algebra of a locally compact

Hausdorff space X that represent a given positive functional π : Cc(X ) → , but there is only one that is a regular Borel measure. In our vector-valued context, we shall have similar results. In this section, we shall deﬁne and investigate the pertinent regularity properties. The general situation is somewhat more involved than the real case.

When X is a locally compact Hausdorff space, then we let B denote its Borel σ-algebra, i.e., B is the σ-algebra that is generated by the open subsets of X .

When E is a σ-monotone complete partially ordered vector space, then we let M(X , B; E+) denote the collection of all positive E-valued measures µ : B → E+.

We distinguish the following regularity properties.

Deﬁnition 3.1. Let X be a locally compact Hausdorff space, let E be a monotone complete partially ordered vector space, and let µ ∈ M(X , B; E+). Then µ is called:

(1) a Borel measure (on X ) when µ(K) ∈ E for all compact subsets K of X ; (2) inner regular at ∆ ∈ B when µ(∆) = {µ(K) : K is compact and K ⊆

∆} in E; (3) weakly inner regular at ∆ ∈ B when µ(∆) = {µ(∆∩K) : K is compact}

in E; (4) outer regular at ∆ ∈ B if µ(∆) = {µ(V ) : V is open and ∆ ⊆ V } in E; (5) a regular Borel measure (on X ) when it is a Borel measure on X that is

inner regular at all open subsets of X and outer regular at all Borel sets; (6) a quasi-regular Borel measure (on X ) when it is a Borel measure on X

that is inner regular at all open subsets of X and weakly inner regular at all Borel sets.

Note that the three subsets of E occurring in the above deﬁnitions are all directed in the appropriate directions, so that the two suprema and the inﬁmum exist in E as a consequence of the monotone completeness of E; σ-monotone completeness would not have sufﬁced here. In the sequel, there will be similar cases where the fact that a set is directed guarantees that a supremum or inﬁmum exists, but we refrain from observing this explicitly each and every time.

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

MARCEL DE JEU

Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands and

Department of Mathematics and Applied Mathematics, University of Pretoria, Corner of Lynnwood Road and Roper Street, Hatﬁeld 0083, Pretoria, South Africa

XINGNI JIANG

College of Mathematics, Sichuan University, No. 24, South Section, First Ring Road, Chengdu, P.R. China

ABSTRACT. We generalise the Riesz representation theorems for positive linear functionals on Cc(X ) and C0(X ), where X is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space E. The representing measures are deﬁned on the Borel σ-algebra of X and take their values in the extended positive cone of E. The corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of X . Results are included where the space E need not be a vector lattice, nor a normed space. Representing measures exist, for example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras.

1. INTRODUCTION AND OVERVIEW

Let X be a locally compact Hausdorff space, and let π : Cc(X ) → Ê be a posit-

ive linear functional. The Riesz representation theorem asserts that there is a unique regular Borel measure on the Borel σ-algebra of X , such that

(1.1)

π( f ) = f dµ

X

E-mail addresses: [email protected], [email protected]

2010 Mathematics Subject Classiﬁcation. Primary 47B65; Secondary 28B15. Key words and phrases. Riesz representation theorem, measure, order integral, ordered vector space.

1

partially

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RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

for all f in Cc(X ). In this paper, we establish analogous representation theorems for positive

linear operators π : Cc(X ) → E and π : C0(X ) → E, where E is a (suitable) partially ordered vector space, while giving explicit formulas for the measures of open and of compact subsets of X .1 Results are included where the space E need not be a vector lattice, nor a normed space. As will become clear, representing measures exist for, e.g., positive linear operators into Banach lattices with order

Ê continuous norms, into the regular operators on KB-spaces, into the self-adjoint

operators on complex Hilbert spaces, and into JBW-algebras. When E = , the results specialise to the classical Riesz representation theorems.

In the general setting of the present paper, the measure µ in equation (1.1) takes its values in the (extended) positive cone of E, and the integral in question is an order integral. Such measures and their order integrals are the subject of [10], which extends earlier work by Wright. The results for the order integral in [10] are fairly complete and also include convergence theorems that can be of use in applications, with the existence theorems in the present paper as a starting point.

A possible multiplicativity of the linear operator π is not an issue in the current paper: being positive and linear is enough. In the sequel [12], we shall consider positive algebra homomorphisms from Cc(X ) or C0(X ) into partially ordered algebras. The representing measures from the current paper can then be shown to be spectral measures that take values in the algebras. It will be seen in [12] that the ensuing existence theorems for abstract spectral measures immediately imply the classical ones for representations of (the complexiﬁcation of) C0(X ) on complex Hilbert spaces, and for positive representations of C0(X ) on KB-spaces in [13]. The up-down theorems that are established in [12] for general partially ordered algebras yield the familiar results for Hilbert spaces as special cases. In [11], which is another sequel to the present paper, we shall be concerned with representation theorems for vector lattices (resp. Banach lattices) of regular operators from Cc(X ) and C0(X ) into Dedekind complete vector lattices (resp. Banach lattices with order continuous norms) in the spirit of [4, Theorem 38.7].

We shall discuss the relation between the present paper and existing representation theorems for positive linear operators in the literature at the end of this introduction. There appears to be no previous work in the vein of the sequels [11] or [12] to the present paper.

This paper is organised as follows. Section 2 contains the necessary prerequisites from [10], including those on

measures and the order integral, and can serve as a summary thereof.

1In the course of the present paper, its prequel [10], and its sequels [11, 12], we shall encounter maps with Cc(X ) or C0(X ) as domains that are sometimes positive linear operators, sometimes vector lattice homomorphisms, and sometimes positive algebra homomorphisms. For each of these contexts, a canonical symbol for such maps could be chosen. However, since our results for these contexts are related, we have chosen to use the same symbol π throughout, thus keeping the notation as uniform as possible.

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

3

In the preparatory Section 3, we introduce various types of regularity of measures on the Borel σ-algebra of a locally compact Hausdorff space that take their values in the extended positive cones of partially ordered vector spaces. A few auxiliary results for the remainder of the paper are also established,

Section 4 contains the proof of a representation theorem (see Theorem 4.2) for positive linear operators π : Cc(X ) → E, where E is (for all practical purposes) a Banach lattice with an order continuous norm. It is one of the essential ingredients for the sequel [11]. The possibly inﬁnite representing measure in it is always regular.

Since spaces of operators will only rarely satisfy the conditions on the space E in Section 4, other results are needed that do apply (at least) when E is a space of operators. Such results are to be found in Section 5, where representation theorems (see Theorems 5.4 and 5.6) are established when the codomain of the positive linear operator π : Cc(X ) → E is a monotone complete and normal space. The class of such spaces E is quite varied. We refer to Examples 2.5 and also to [10, Section 3] for examples, some of which were already mentioned above. When E consists of the regular operators on a Banach lattice with an order continuous norm, or of the self-adjoint operators in a strongly closed complex linear subspace of the bounded linear operators on a complex Hilbert space, then the representing measure in Theorem 5.4 has the familiar property of being σ-additive in the strong operator topology. The representing measures in Theorem 5.4 are ﬁnite (by assumption) and regular. Those in Theorem 5.6 can be inﬁnite, but regularity need then hold only locally. In Remark 5.8, we compare the applicability of the main representation theorems (Theorems 4.2, 5.4, and 5.6) for positive linear operators π : C0(X ) → E.

The ﬁnal Section 6 is concerned with positive linear operators π : C0(X ) → E. The domain C0(X ) is now a Banach lattice, and we can then exploit automatic continuity to derive representation theorems for such linear operators from those for their restrictions to Cc(X ). The representing measures thus obtained are all ﬁnite. Theorem 6.2 covers the case where E is a KB-space, and Theorem 6.8 applies to the larger class of quasi-perfect partially ordered vector spaces. In Theorems 6.10 and 6.12, the space E consists of the regular operators on a KB-space and of the self-adjoint operators in a strongly closed complex linear subspace of the bounded linear operators on a complex Hilbert space, respectively. In both cases, the representing measures are strongly σ-additive again. Furthermore, a number of SOT-closed subspaces of E that are naturally associated with π can be seen to coincide. As a consequence, the representing measure takes its values in the coinciding bicommutants of π(Cc(X )) and π(C0(X )).

We now give an overview of earlier work that we are aware of on Riesz representation theorems for positive linear operators from spaces of continuous functions into various types of partially ordered vector spaces. The space X is always a locally compact Hausdorff space, unless otherwise stated. As will become clear,

4

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

one can hardly speak of ‘the’ Riesz representation theorem for positive linear operators from Cc(X ) or C0(X ) into partially ordered vector spaces.

To start with, there is the seminal paper by Wright [23]. It has a representation theorem for a positive linear operator from C(X ), where X is compact, into a Stone algebra. A second paper [24] by the same author covers the case of a positive linear operator from C(X ), where X is compact, into a σ-Dedekind complete vector lattice. A third paper [25] contains a representation theorem for a positive linear operator from Cc(X ) into a Dedekind complete vector lattice. In a fourth [26], the existence is established of a representing Baire measure for a positive linear operator from C(X ), where X is compact, into a σ-monotone complete partially ordered vector space. When comparing the latter result to our Theorem 5.4, both have their strong points. Wright’s conditions on E are more lenient, but the space X has to be compact, which is essential to the proofs. Theorem 5.4, on the other hand, is valid for locally compact spaces, and also gives explicit formulas for the measure of open and of compact subsets.

In [16], one of Khurana’s papers in this direction, it is shown that a positive linear operator from Cc(X ) into a monotone complete partially ordered vector space can be extended to a σ-order continuous linear operator deﬁned on the bounded Borel functions with compact support; this goes in the direction of a Riesz representation theorem. In a second paper [17], a representation theorem is established, under certain additional conditions, for a positive linear operator from the bounded continuous functions on a completely regular Hausdorff space into a monotone complete partially ordered vector space. In [18], he proves a representation theorem for a positive linear operator from the continuous functions on a completely regular T1-space into a Stone algebra.

In [19], Lipecki shows the existence of a representing ﬁnitely additive measure for positive linear operators from the bounded continuous functions on an arbitrary topological space into a monotone complete partially ordered vector space; there is a second result for the bounded continuous functions when the topological space is normal.

In [8], Coquand gives a new proof of Wright’s result in [26]. The results cited above also include regularity properties of the representing measures, but not always the same properties. In cases where several results apply, it is, therefore, not clear whether the representing measures are necessarily equal. The results in the present paper have the advantage that the space X need not be compact and that they apply, amongst others, when E is a monotone complete normal space. This is a fairly large class of spaces, containing many spaces that are not vector lattices. Moreover, explicit formulas for the measures of open and of compact subsets of X are given. We are not aware of a similar combination of a reasonably wide range of applicability and concrete formulas in the literature on partially ordered vector spaces. The usefulness of this combination can be seen in, e.g, Theorems 6.10 and 6.12, which, though in two rather different contexts, are both virtually immediate consequences of one underlying general result. It will become even more pronounced in the sequel [12]

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

5

when considering positive algebra homomorphisms from Cc(X ) and C0(X ) into monotone complete normal partially ordered algebras with monotone continu-

ous multiplications.

2. PRELIMINARIES

In this section, we collect some conventions, deﬁnition, notations, and preparatory results that will be used in the sequel.

The indicator function of a subset S of a set X is denoted by χS. We shall also write 0 for χ and 1 for χX . When X is a topological space, then we let Cc(X ), resp. C0(X ), denote the continuous real-valued functions on X that have compact support, resp. vanish at inﬁnity.

2.1. Partially ordered vector spaces. All vector spaces we shall consider are over the real numbers, unless otherwise indicated. An operator between two vector spaces and a functional are always supposed to be linear, but—when this notion is applicable—need not be bounded. We do not require that the positive cone E+ of a partially ordered vector space E be generating. Equivalently, we do not require that E be directed. We do require, however, that E+ be proper, i.e., that E+ ∩ (−E+) = {0}. All vector lattices are supposed to be Archimedean.

Deﬁnition 2.1. A partially ordered vector space E is called (1) σ-monotone complete if every increasing sequence {xn}∞ n=1 in E that is bounded from above has a supremum in E; (2) monotone complete if every increasing net {xλ}λ∈Λ in E that is bounded from above has a supremum in E; (3) σ-Dedekind complete if every non-empty at most countably inﬁnite subset S of E that is bounded from above has a supremum in E; (4) Dedekind complete if every non-empty subset S that is bounded from above has a supremum in E.

We shall employ the usual notation in which xλ ↓ means that {xλ}λ∈Λ is a decreasing net, and in which xλ ↓ x means that {xλ}λ∈Λ is a decreasing net with inﬁmum x. The notations xλ ↑ and xλ ↑ x are similarly deﬁned.

It was observed in [26, Lemma 1.1] that every σ-monotone complete partially ordered vector space E (and then also every monotone complete, σ-Dedekind complete, or Dedekind complete partially order vector space) is Archimedean, i.e., {ǫ x : ǫ > 0} = 0 for all x ∈ E+. We shall use this a number of times.

Vector spaces of operators between partially ordered vector spaces can inherit completeness properties from the codomains. In order to formulate this, we ﬁrst introduce some notation and terminology.

When E and F are vector spaces, then L(E, F) denotes the vector space of operators from E into F. An operator T ∈ L(E, F) between two partially ordered vector spaces is positive if T (E+) ⊆ F+, and regular if it is the difference of two positive operators. The regular operators from E into F form a vector space that is denoted by Lr(E, F). When E+ is directed, then every linear subspace

6

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

Ê of L(E, F) that contains Lr(E, F) is naturally partially ordered with the regular

positive operators from E into F, denoted by Lr(E, F)+, as its positive cone. We shall write L(E) for L(E, E), Lr(E) for Lr(E, E), and E for Lr(E, ). When E is a Banach lattice, then E coincides with the norm dual E∗ of E.

We can now state how completeness is hereditary. When E is a directed partially ordered vector space, and F is a partially ordered vector space that is monotone complete (resp. σ-monotone complete), then any linear subspace of L(E, F) containing Lr(E, F) is monotone complete (resp. σ-monotone complete); see [10, Proposition 3.1].

The monotone complete partially ordered vector spaces that are also normal will play an important part in this paper. We now proceed to deﬁne the latter notion.

Deﬁnition 2.2. Let E and F be partially ordered vector spaces, and let T : E → F be a positive operator. Then T is called order continuous (resp. σ-order continuous) if T xλ ↓ 0 in F whenever xλ ↓ 0 in E (resp. if T xn ↓ 0 in F whenever xn ↓ 0 in E). A general operator in Lr(E, F) is called order continuous (resp. σ-order continuous) if it is the difference of two positive order continuous operators. We let Loc(E, F ) (resp. Lσoc(E, F )) denote the order continuous (resp. σ-order

Ê continuous) operators from E into F; we shall write Eoc for Loc(E, ) and Eσoc

for Lσoc(E, ).2

It is easy to see that Loc(E, F ) and Lσoc(E, F ) are linear subspaces of Lr(E, F ). When E is directed, then they are partially ordered vector spaces with the positive order continuous operators (resp. the positive σ-order continuous operators) from E into F as positive cones, which are generating by deﬁnition.

Deﬁnition 2.3. Let E be a partially ordered vector space. Then E is called normal when, for x ∈ E, (x, x′) ≥ 0 for all x′ ∈ (Eoc)+ if and only if x ∈ E+. We say that E is σ-normal when, for x ∈ E, (x, x′) ≥ 0 for all x′ ∈ (Eσoc)+ if and only if x ∈ E+.3

The importance of normality for our work lies in the following result (see [10, Proposition 3.8]) that will be used quite a few times in the present paper.

Proposition 2.4. Let E be a normal partially ordered vector space. Suppose that {xλ}λ∈Λ is a net in E, and that x ∈ E.

(1) If xλ ↓, then xλ ↓ x if and only if (x , x ′) = infλ∈Λ(xλ, x ′) for all x ′ ∈ (Eoc)+.

2When E and F are vector lattices, where F is Dedekind complete, then the notions of order continuous and σ-order continuous operators in Deﬁnition 2.2 agree with the usual ones in the literature as on [27, p. 123]; see [10, Remark 3.5] for this.

3When E is a vector lattice, then the notion of normality in Deﬁnition 2.3 coincides with the usual one in the literature (see [1, p. 21], for example) that Eoc separates the points of E; see [10, Lemma 3.7] for this. It also follows from [10, Lemma 3.7] that a vector lattice E is σ-normal as in Deﬁnition 2.3 if and only if Eσoc separates the points of E.

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

7

(2) If xλ ↑, then xλ ↑ x if and only if (x , x ′) = supλ∈Λ(xλ, x ′) for all x ′ ∈ (Eoc)+.

In the presence of monotone completeness, normality of a codomain is a hereditary property: when E is a directed partially ordered vector space, and F is a monotone complete and normal partially ordered vector space, then any linear subspace of L(E, F) containing Lr(E, F) is monotone complete and normal; see [10, Proposition 3.11].

Since both (σ-)monotone completeness and normality are properties that are inherited by partially ordered vector spaces of operators, it is easy to construct examples of monotone complete and normal spaces once one has such a space to begin with.

Examples 2.5. In [10, Section 3] we have included a number of examples of partially ordered vector spaces that are monotone complete and normal. These include (but are not limited to):

(1) Banach lattices with order continuous norms; (2) for partially ordered vector spaces E and F such that E is directed and

F is monotone complete and normal: every linear subspace of L(E, F) that contains Lr(E, F); (3) as a special case of part (2): the regular operators on a Banach lattice with an order continuous norm; (4) the real vector space that consists of the self-adjoint operators in a strongly closed complex linear subspace of the bounded operators on a complex Hilbert space; (5) JBW-algebras.4

2.2. Measures and order integrals. In this section, we shall brieﬂy summarise the relevant deﬁnitions and results for measures and order integrals from [10, Sections 4 and 6]. This extends earlier work by Wright and contains the usual Lebesgue integral as a special case. We refer to [10] for a discussion of the relation with Wright’s work.

We refrain from mentioning here in any detail the material on outer measures in [10, Section 5]. It is indispensable in the proof of Theorem 4.2, but it occurs only as an intermediate step and does not reappear.

Let E be a partially ordered vector space. As the case of the Lebesgue integral on the real line already shows, one cannot expect a representing measure for a positive operator π : Cc(X ) → E to be ﬁnite. This is why we extend E by introducing the set E := E ∪ {∞} as a disjoint union, and extend the partial ordering from E to E by declaring that x ≤ ∞ for all x ∈ E. Then E+ := E+ ∪ {∞} is the set of positive elements of E. The elements of E that are in E are called ﬁnite. One makes E into an abelian monoid by retaining the addition on E and deﬁning ∞ + x := ∞ and x + ∞ := ∞ for all x ∈ E. Then E+ is a

4We shall use [2,3] references for JBW-algebras. In these books, a JBW-algebra is supposed to have an identity element; see [3, Deﬁnitions 1.5 and 2.2]. In other sources, this is not supposed. However, as [15, Lemma 4.1.7] shows, the existence of an identity element is, in fact, automatic.

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RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

sub-monoid of E. When x, y ∈ E are such that x ≤ y, then x + z ≤ y + z for all

Ê Ê z ∈ E. We keep the action of on E, and deﬁne r ·∞ := ∞ for all r ∈ + \{0} Ê and 0 · ∞ := 0. Then r x ≤ r y for all r, s ∈ + and x, y ∈ E such that r ≤ s and

x ≤ y. Furthermore, r(x + y) = r x + r y, (r + s)x = r x + sx, and (rs)x = r(sx)

Ê for all r, s ∈ + and x, y ∈ E+. Such relations will be used in the sequel without

further reference. When being used to working with the extended real numbers, which are

still linearly ordered, it may be relatively easy to make mistakes when arguing

with the ordering of and the operations on E and E. It is for this reason that a fair number of technical tools have been collected in [10, Lemmas 2.3 to 2.5] that will be used repeatedly in the present paper and its sequels. Whenever

necessary, we shall be careful to indicate whether we are working in E or in E when speaking of order bounds, suprema, or inﬁma.

Now that we have the extended space E available, it is possible to deﬁne the measures that concern us. It was Wright who ﬁrst observed in [23] that equation (2.1), below, is the proper way to generalise the notion of σ-additivity from the real numbers to more general partially ordered vector spaces.

A measurable space is a pair (X , Ω), where X is a set and Ω is an algebra of subsets of X ; i.e., Ω is a non-empty collection of subsets of X that is closed under taking complements and under taking ﬁnite unions. For the moment, we can still work with algebras of subsets, rather than σ-algebras.

Deﬁnition 2.6. Let (X , Ω) be a measurable space, and let E be a σ-monotone

complete partially ordered vector space. A positive E-valued measure is a map

µ : Ω → E+ such that:

(1) µ( ) = 0;

(2) (σ-additivity) whenever {∆n}∞ n=1 is a pairwise disjoint sequence in Ω

with

∞ n=1

∆n

∈

Ω,

then

∞

∞N

(2.1)

µ

∆n =

µ(∆n)

n=1

N =1 n=1

in E.

Since µ is E+-valued, it follows from the σ-monotone completeness of E that

the supremum in the right hand side of equation (2.1) exists in E; see part (1) of [10, Lemma 2.5]. When µ(X ) ∈ E+, then we say that µ is ﬁnite, or that it

Ê is E-valued; when µ(X ) = ∞, then µ is said to be inﬁnite. As [10, Section 4]

shows, a good number of the properties of positive -valued measures hold in the general case as well, including even the Borel–Cantelli lemmas (see [10, Lemma 4.7]).

Next, we introduce the integral with respect to a measure. From now on, we suppose that Ω is a σ-algebra.

Ê A measurable function ϕ : X → + is an elementary function if it takes only

ﬁnitely many (ﬁnite) values. It can be written as a ﬁnite sum ϕ = ni=1 riχ∆i

Ê for some r1, . . . , rn ∈ + and ∆1, . . . , ∆n ∈ Ω. Here the ri are all ﬁnite, but it

RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

9

is allowed that µ(∆i) = ∞ for some of the ∆i. We deﬁne its (order) integral, which is an element of E+, by setting

o

n

ϕ dµ := riµ(∆i).

X

i=1

This deﬁnition is independent of the above expression for ϕ as a ﬁnite sum.

Ê When f : X → + is measurable, we choose a sequence {ϕn}∞ n=1 of elementary Ê functions such that ϕn ↑ f pointwise in +. We deﬁne the order integral of f ,

which is an element of E+, by

o

∞

f dµ :=

X

n=1

o

ϕn dµ.

X

Since Xoϕn dµ ↑, the σ-monotone completeness of E guarantees that this supremum exists in E+. It is independent of the choice of the sequence {ϕn}∞ n=1.

Ê We let L 1(X , Ω, µ; ) denote the set of all (ﬁnite-valued) measurable funcÊ tions f : X → such that o| f | dµ is ﬁnite. By splitting a function into its pos-

X

Ê itive and negative parts, the order integral is then deﬁned on L 1(X , Ω, µ; ). Ê We let B(X , Ω; ) denote the bounded measurable functions on X . For a ﬁnite Ê Ê measure µ, B(X , Ω; ) ⊆ L 1(X , Ω, µ; ).

In [10, Section 6.2], the monotone convergence theorem for the order integ-

ral is established. When E is σ-Dedekind complete, then Fatou’s lemma and the

dominated convergence theorem are also valid.

Ê The space L 1(X , Ω, µ; ) is a σ-Dedekind complete vector lattice, and the Ê order integral is a σ-order continuous positive operator from L 1(X , Ω, µ; ) Ê Ê into E; see [10, Proposition 6.14]. According to [10, Theorem 6.17], the space

L1(X , Ω, µ; ), where elements of L 1(X , Ω, µ; ) have been identiﬁed when

they agree µ-almost everywhere, is likewise a σ-Dedekind complete vector lat-

tice, and the order integral induces a strictly positive σ-order continuous op-

Ê erator Iµ from L1(X , Ω, µ; ) into E. When E is monotone complete and has Ê the countable sup property, then L1(X , Ω, µ; ) is a Dedekind complete vector

lattice with the countable sup property, and Iµ is order continuous.5

Remark 2.7.

(1) When E is a partially ordered Banach space with a closed positive cone, then every positive vector measure is a measure in the sense of Deﬁnition 2.6, but not conversely. Even when a measure falls into both categories, the domain of the order integral can properly contain that of

5As in [10, Section 6], we say that a partially ordered vector space E has the countable sup

property when, for every net {xλ}λ∈Λ ⊆ E+ and x ∈ E+ such that xλ ↑ x, there exists an at most countably inﬁnite set of indices {λn : n ≥ 1} such that x = supn≥1 xλn . In this case, there also always exist indices λ1 ≤ λ2 ≤ · · · such that xλn ↑ x. For vector lattices, this deﬁnition coincides with the usual one as can be found on, e.g., [5, p. 103].

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RIESZ REPRESENTATION THEOREMS FOR POSITIVE LINEAR OPERATORS

any reasonably deﬁned integral with respect to the vector measure using Banach space methods. We refer to [10, Section 7] for this discussion. (2) When E consists of the regular operators on a Banach lattice with an order continuous norm, or when E consists of the self-adjoint operators in a strongly closed complex linear subspace of the bounded operator on a complex Hilbert space, then the ﬁnite measures in the sense of Deﬁnition 2.6 are precisely the set maps µ : Ω → E+ with µ( ) = 0 that are σ-additive with respect to the strong operator topology on E; see [10, Lemmas 4.2 and 4.3].

3. MEASURES ON LOCALLY COMPACT HAUSDORFF SPACES

Ê There may be several measures on the Borel σ-algebra of a locally compact

Hausdorff space X that represent a given positive functional π : Cc(X ) → , but there is only one that is a regular Borel measure. In our vector-valued context, we shall have similar results. In this section, we shall deﬁne and investigate the pertinent regularity properties. The general situation is somewhat more involved than the real case.

When X is a locally compact Hausdorff space, then we let B denote its Borel σ-algebra, i.e., B is the σ-algebra that is generated by the open subsets of X .

When E is a σ-monotone complete partially ordered vector space, then we let M(X , B; E+) denote the collection of all positive E-valued measures µ : B → E+.

We distinguish the following regularity properties.

Deﬁnition 3.1. Let X be a locally compact Hausdorff space, let E be a monotone complete partially ordered vector space, and let µ ∈ M(X , B; E+). Then µ is called:

(1) a Borel measure (on X ) when µ(K) ∈ E for all compact subsets K of X ; (2) inner regular at ∆ ∈ B when µ(∆) = {µ(K) : K is compact and K ⊆

∆} in E; (3) weakly inner regular at ∆ ∈ B when µ(∆) = {µ(∆∩K) : K is compact}

in E; (4) outer regular at ∆ ∈ B if µ(∆) = {µ(V ) : V is open and ∆ ⊆ V } in E; (5) a regular Borel measure (on X ) when it is a Borel measure on X that is

inner regular at all open subsets of X and outer regular at all Borel sets; (6) a quasi-regular Borel measure (on X ) when it is a Borel measure on X

that is inner regular at all open subsets of X and weakly inner regular at all Borel sets.

Note that the three subsets of E occurring in the above deﬁnitions are all directed in the appropriate directions, so that the two suprema and the inﬁmum exist in E as a consequence of the monotone completeness of E; σ-monotone completeness would not have sufﬁced here. In the sequel, there will be similar cases where the fact that a set is directed guarantees that a supremum or inﬁmum exists, but we refrain from observing this explicitly each and every time.