Category measures, the dual of C(K)^δ and hyper-Stonean spaces
aa r X i v : . [ m a t h . F A ] F e b CATEGORY MEASURES, THE DUAL OF C( K ) δ ANDHYPER-STONEAN SPACES
JAN HARM VAN DER WALT
Abstract.
For a compact Hausdorff space K , we give descriptions of thedual of C( K ) δ , the Dedekind completion of the Banach lattice C( K ) of con-tinuous, real-valued functions on K . We characterize those functionals whichare σ -order continuous and order continuous, respectively, in terms of Ox-toby’s category measures. This leads to a purely topological characterizationof hyper-Stonean spaces. Introduction
It is well known that the Banach lattice C( K ) of continuous, real-valued functionson a compact Hausdorff space K fails, in general, to be Dedekind complete. In fact,C( K ) is Dedekind complete if and only if K is Stonean, i.e. the closure of every openset is open, see for instance [25]. A natural question is to describe the Dedekindcompletion of C( K ); that is, the unique (up to linear lattice isometry) Dedekindcomplete Banach lattice containing C( K ) as a majorizing, order dense sublattice.Several answers to this question have been given, see for instance [3, 8, 10, 14, 21,22, 27] among others. A further question arises: what is the dual of C( K ) δ ? As faras we are aware, no direct answer to this question has been stated explicitly in theliterature.We give two answers to this question; one in terms of Borel measures on theGleason cover of K [13], and another in terms of measures on the category algebraof K [23, 24]. We use the latter description to obtain natural characterizations ofstrictly positive, σ -order continuous, and order continuous functionals on C( K ) δ .In particular, we show that C( K ) δ admits a strictly positive, σ -order continuousfunctional if and only if K admits a category measure in the sense of Oxtoby[23]. Specialising to the case of a Stonean space K , we obtain a purely topologicalcharacterization of hyper-Stonean spaces. Although the class of hyper-Stoneanspaces was introduced by Dixmier [11] in 1951, no such characterization has beengiven to date [7, page 197].The paper is organised as follows. In Section 2 we introduce notation and recalldefinitions and results from the literature to be used throughout the rest of thepaper. Section 3 contains two representations of C( K ) δ which are amendable to theproblem of characterizing the dual of this space; these characterizations of (C( K ) δ ) ∗ are given in Section 4. Strictly positive, σ -order continuous and order continuousfunctionals on C( K ) δ are discussed in Section 5 which ends with a characterizationof hyper-Stonean spaces. Mathematics Subject Classification.
Primary 46E05, 54G05; Secondary 46E27, 54F65.
Key words and phrases.
Hyper-Stonean spaces, category measures, Banach lattices, continuousfunctions. Preliminaries
Vector and Banach lattices.
Let E be a real vector lattice. We assumethat E is Archimedean; that is, inf { n f : n ∈ N } = 0 for every f ≥
0. Denote by E + the positive cone in E , E + = { f ∈ E : 0 ≤ f } . For f ∈ E the positive part,negative part and modulus of f is given by f + = f ∧ , f − = ( − f ) ∧ | f | = f + + f − , respectively. For D ⊆ E and f ∈ E we write D ↓ f if D is downward directed andinf D = f . E is Dedekind complete if every subset of E which is bounded above (below)has a least upper bound (greatest lower bound). We call E order separable if everysubset A of E contains a countable set with the same set of upper bounds as A .Let F be a vector lattice subspace of E . F is order dense in E if for every 0
Let E be an Archimedean vector lattice and F an order dense, ma-jorizing vector lattice subspace of E . (i) If ϕ ∈ E ∼ then ϕ is order continuous on E if and only if the restriction of ϕ to F is order continuous. (ii) If ϕ ∈ E ∼ is σ -order then the restriction of ϕ to F is σ -order continuous. (iii) If ϕ ∈ F ∼ is order continuous, then ϕ has a unique order continuous ex-tension to E .Proof. We prove the results for positive functionals. The case of a general ϕ ∈ E ∼ follows from the decomposition ϕ = ϕ + − ϕ − . ATEGORY MEASURES, THE DUAL OF C( K ) δ AND HYPER-STONEAN SPACES 3
Proof of (i).
Let D ⊆ F + . Because F is order dense in E , D ↓ E if and onlyif D ↓ F . Therefore, if ϕ is order continuous on E then its restriction to F isorder continuous.Conversely, assume that the restriction of ϕ to F is order continuous. Let D ↓ E . For each g ∈ D let D g = { f ∈ F : g ≤ f } , and let D ′ = S { D g : g ∈ D } .The fact that F is majorizing in E guarantees that D g = ∅ for each g ∈ D . Weobserve that D ↓ E implies that D ′ is downward directed. In particular, if f, h ∈ D ′ then f ∧ h ∈ D ′ . Because F is an order dense sublattice of E , g = inf D g for every g ∈ D . Therefore D ′ ↓ E , hence also in F . It follows from thepositivity and order continuity of ϕ on F that 0 ≤ inf ϕ [ D ] ≤ inf ϕ [ D ′ ] = 0.Therefore inf ϕ [ D ] = 0. Proof of (ii).
The proof is similar to the proof of (i).
Proof of (iii).
This is a special case of [2, Theorem 4.12]. (cid:3) K ) and its dual. Let K be a compact Hausdorff space. C( K ) is the realBanach lattice of continuous, real-valued functions on K ; the algebraic operationsand order relation are defined pointwise as always, and the norm on C( K ) is k f k =max { f ( x ) | : x ∈ K } . The symbols and denote the constant functions withvalues 0 and 1, respectively. For A ⊆ K , the characteristic function of A is denoted A .The space K satisfies the countable chain condition (ccc) if every pairwise disjointcollection of nonempty open sets in K is countable. C( K ) is order separable if andonly if K satisfies ccc [9, Theorem 10.3].Σ B K denotes the σ -algebra of Borel sets in K and M( K ) the space of regularsigned Borel measures on K ; that is, σ -additive, real valued regular measures onΣ B K . As is well known, M( K ) is a Banach lattice with respect to the total variationnorm k ϕ k = | ϕ | ( K ) , ϕ ∈ M( K )and pointwise ordering. In fact, M( K ) is isometrically lattice isomorphic to C( K ) ∗ .We will not make a notational distinction between a measure ϕ and the associatedfunctional. A measure ϕ ∈ M( K ) is called normal [7, 11] if the associated functionalon C( K ) is order continuous. We denote the set of normal measures on by N( K ).Normal measures can be characterised as follows: ϕ ∈ M( K ) is normal if and onlyif | ϕ | ( N ) = 0 for every meagre set N ∈ Σ B K , see for instance [7, Theorem 7.4.7].Recall that K is Stonean if the closure of every open set is open. As mentionedin the introduction, K is Stonean if and only if C( K ) is Dedekind complete. If K is Stonean and, in addition, the union of the suppports of the normal measures on K is dense in K , we call K hyper-Stonean , see [11] and [7, Section 4.7]. We recallthe following result which contains the main motivation for studying hyper-Stoneanspaces, see for instance [7]. Theorem 2.2.
A space K is hyper-Stonean if and only if any one of the followingconditions hold. (i) For every < f ∈ C( K ) there exists a positive order continuous functional ϕ on C( K ) so that ϕ ( f ) > . (ii) C( K ) is isometrically isomorphic to the dual of a Banach space. (iii) C( K ) is isometrically lattice isomorphic to the dual of a Banach lattice. (iv) The C ∗ -algebra of complex-valued continuous functions on K is a von Neu-mann algebra. JAN HARM VAN DER WALT
Category measures.
Recall that a subset A of K has the property of Baireif there exists an open set U and a meagre set N in K so that A = U ∆ N . Let N K = { N ⊂ K : N is meagre } , Σ C K = { U ∆ N : U is open in K, N ∈ N } . (2.1)The collection Σ C K is a σ -algebra [24, Theorrem 4.3] and N K is a σ -ideal in Σ C K .In fact, N K is a σ -ideal in the powerset of K . The quotient algebra Σ C K / N K isa complete Boolean algebra. In fact, Σ C K / N K is isomorphic to the algebra R K ifregular open subsets of K , see [21, 23].A category measure on K is positive, σ -additive measure µ : Σ C K → R with theproperty that for any A ∈ Σ C K , µ ( A ) = 0 if and only if A ∈ N K , see for instance [23].There is a bijective correspondence between category measures on K and strictlypositive, σ -additive measures on R K . Oxtoby [23, 24] investigated the problem ofcharacterizing those topological spaces which admit a category measure. A solutionto this problem was given by J. M. Ayerbe Toledano [26] . In order to formulatethis result we recall the following definitions, see [4, 16, 26]. Definition 2.3.
Let X be a nonempty set and F a collection of nonempty subsetsof X . For any n ∈ N and ¯ F = h F , . . . , F n i ∈ F n let i ( ¯ F ) = max {| J | : J ⊆ { , . . . n } , \ j ∈ J F j = ∅} . The Kelley intersection number of F is defined as k ( F ) = inf { i ( ¯ F ) n : n ∈ N , ¯ F ∈ F n } . Definition 2.4.
Let K be a compact Hausdorff space and T the collection ofnonempty, open subsets of K . We say that K satisfies property (***) if there existsa partition {T n : n ∈ N } of T such that the following conditions are satisfied forevery n ∈ N .(i) k ( T n ) > U m ) is an increasing sequence of open sets so that S { U m : m ∈ N } ∈ T n then there exists m ∈ N so that U m ∈ T n .(iii) If U ∈ T n and V is an open set so that U ∆ V is meagre then V ∈ T n .Ryll-Nardzewski obtained a necessary and sufficient condition of the existenceof a σ -additive, strictly positive measure on a complete Boolean algebra, see [16,Adendum]. It is easy to see that property (***) is equivalent to R K satisfying Ryll-Nardzewski’s condition. The following theorem of Ayerbe Toledano [26] thereforefollows immediately from Ryll-Nardzewski’s result. Theorem 2.5.
A compact Hausdorff space K admits a category measure if andonly if it satisfies property (***).Remark . Cambern and Greim [6], see also [7], define a category measure tobe a (not necessarily finite) positive Borel measure µ on a Stonean space K whichsatisfies the following conditions:(i) µ is regular on closed sets of finite measure,(ii) µ ( N ) = 0 for every meagre Borel set N , and, We formulate the result for compact Hausdorff spaces, but it is valid for arbitrary Bairespaces.
ATEGORY MEASURES, THE DUAL OF C( K ) δ AND HYPER-STONEAN SPACES 5 (iii) for every nonempty clopen set A there exists a clopen set A ⊆ A so that0 < µ ( A ) < ∞ .Such measures are called perfect in [5], and are distinct from the category measuresdiscussed here. 3. Characterizations of C( K ) δ The Gleason cover and the Maeda-Ogasawara theorem.
Let K be acompact Hausdorff space. A seminal result of Gleason [13] associates, in a canonicalway, with K a Stonean space G K , its projective cover. In order to formulatethis result we recall the following. If L is compact Hausdorff space, a continuoussurjection f : K → L is called irreducible if f [ C ] = L for every proper closed subset C of K . Theorem 3.1.
Let K be a compact Hausdorff space. There exists a Stonean space G K , unique up to homeomorphism, and an irreducible map π K : G K → K . Because π K is onto, the induced map T π K : C( K ) ∋ f f ◦ π K ∈ C( G K ) is anisometric linear lattice isomorphism onto a closed vector lattice subspace of C( G K ).Moreover, since π K is irreducible, π ∗ K [C( K )] is a order dense sublattice of C( G K ).Clearly, π ∗ K [C( K )] is majorising in C( G K ), seeing as it contains the order unit G K of C( G K ). Lastly we note that, G K being Stonean, C( G K ) is Dedekind complete.Hence we have the following. Theorem 3.2.
Let K be a compact Hausdorff space. Then C( K ) δ is isometricallylattice isomorphic to C( G K ) . Let ¯ R be the two-point compactification of R . Denote by C ∞ ( K ) the space ofcontinuous functions f : K → ¯ R so that f − [ R ] is dense (hence open and dense) in K . If K is a Stonean space then C ∞ ( K ) is a universally complete vector lattice,see for instance [1, Theorem 7.27], and C( K ) is the ideal in C ∞ ( K ) generated by . A classical result in the representation theory for vector lattices is due to Maedaand Ogasawara [20], see for instance [1] for a more recent presentation. Theorem 3.3.
Let E be an Archimedean vector lattice with weak order unit e .There exists a Stonean space Ω E and a linear lattice isomorphism T : E → C ∞ (Ω E ) onto an order dense sublattice of C ∞ (Ω E ) so that T e = . In Theorem 3.3, let E = C( K ). Then T ( K ) = Ω E . For f ∈ C ∞ (Ω E ), f ∈ C(Ω E ) if and only if there exists c > | f | ≤ c Ω E . Therefore T [C( K )] isan order dense and majorizing vector lattice subspace of C ∞ (Ω E ). Hence we havethe following. Theorem 3.4.
Let K be a compact Hausdorff space. Then C( K ) δ is isometricallylattice isomorphic to C(Ω C( K ) ) . A compact Hausdorff space is uniquely determined, up to homeomorphism, byits lattice of real-valued continuous functions. The Dedekind completion of a vectorlattice is unique up to a linear lattice isomorphism. Therefore Ω C( K ) and G K arehomeomorphic. This can be seen directly by recalling the constructions of Ω C( K ) and G K , respectively. G K may be constructed as the Stone space of R K . On theother hand, Ω C( K ) is the Stone space of the Boolean algebra of bands in C( K )which is isomorphic to R K . JAN HARM VAN DER WALT
Category measurable functions.
Denote by B(Σ C K ) the Archimedean vec-tor lattice consisting of all real-valued, bounded and Σ C K -measurable functions on K . The subset N(Σ C K , N K ) = { f ∈ B(Σ C K ) : K \ f − [ { } ] ∈ N K } of B(Σ C K ) is a σ -ideal in B(Σ C K ). ThereforeL ∞ (Σ C K , N K ) = B(Σ C K ) / N(Σ C K , N K )is an Archimedean vector lattice [19, Theorem 60.3]. For each f ∈ B(Σ C K ) wedenote by ˆ f the equivalence class in L ∞ (Σ C K , N K ) generated by f . We note thatfor ˆ f , ˆ g ∈ L ∞ (Σ C K , N K ),ˆ f ≤ ˆ g if and only if { x ∈ K : f ( x ) > g ( x ) } ∈ N K . (3.1)For ˆ f ∈ L ∞ (Σ C K , N K ) set k ˆ f k ∞ = inf { c ≥ | ˆ f | ≤ c ˆ } . Note that for ˆ f ∈ L ∞ (Σ C K , N K ), k ˆ f k ∞ = inf { c ≥ { x ∈ K : | f ( x ) | > c } ∈ N K } . With respect to this norm the vector lattice L ∞ (Σ C K , N K ) is a Banach lattice. Be-cause Σ C K / N K is complete, L ∞ (Σ C K , N K ) is Dedekind complete, see for instance[12, 363M & 363N]. We remark that it is possible to prove this result directly.Every equivalence class in L ∞ (Σ C K , N K ) contains a bounded lower semi-continuousfunction, and, every bounded lower semi-continuous function belongs to B(Σ C K ).Dedekind completeness of L ∞ (Σ C K , N K ) then follows from the fact that the point-wise supremum of any family of lower semi-continuous functions is again lowersemi-continuous. Theorem 3.5.
Let K be a compact Hausdorff space. Then C( K ) δ is isometricallylattice isomorphic to L ∞ (Σ C K , N K ) .Proof. C( K ) is vector lattice subspace of B(Σ C K ). Furthermore, for C( K ) ∩ N(Σ C K , N K ) = { } . Therefore the mapping T : C( K ) ∋ u ˆ u ∈ L ∞ (Σ C K , N K ) is a linear latticeisomorphism onto a vector lattice subspace of L ∞ (Σ C K , N K ). Clearly T is an iso-metry and T [C( K )] is majorising in L ∞ (Σ C K , N K ).Since L ∞ (Σ C K , N K ) is a Dedekind complete Banach lattice, it remains only toverify that T [C( K )] is order dense in L ∞ (Σ C K , N K ). To this end, consider ˆ < ˆ u ∈ L ∞ (Σ C K , N K ). There exists a real number ǫ >
0, an open set U in K and N ∈ N K so that ǫ < u ( x ) for every x ∈ U ∆ N . According to (3.1) there exists M ∈ N K so that 0 ≤ u ( x ) for all x ∈ K \ M . Consider a function v ∈ C( K ) so that < v ≤ ǫ U . Fix x ∈ K \ ( M ∪ N ). If x ∈ U then v ( x ) ≤ ǫ < u ( x ). If x ∈ K \ U then v ( x ) = 0 ≤ u ( x ). Therefore v ( x ) ≤ u ( x ) for all x ∈ K \ ( M ∪ N ) so that T v = ˆ v ≤ ˆ u by (3.1). (cid:3) The dual of C( K ) δ From the results discussed in Section 3, in particular Theorems 3.4 and 3.5, weobtain immediately two characterizations of the dual of C( K ) δ . The first followsdirectly from the Riesz Representation Theorem for compact Hausdorff spaces. ATEGORY MEASURES, THE DUAL OF C( K ) δ AND HYPER-STONEAN SPACES 7
Theorem 4.1.
Let K be a compact Hausdorff space. Then (C( K ) δ ) ∗ is isometric-ally lattice isomorphic to M( G K ) . Theorem 3.5 yields a second characterization of (C( K ) δ ) ∗ . Denote by M(Σ C K , N K )the set of bounded, finitely additive signed measures µ on Σ C K with the propertythat µ ( N ) = 0 for every N ∈ N K . The space M(Σ C K , N K ) is a vector lattice [28,Section 1]. With respect to the total variation norm k ϕ k = | ϕ | ( K ) , ϕ ∈ M(Σ C K , N K )M(Σ C K , N K ) is a Banach lattice, see for instance [18]. We now have the following. Theorem 4.2.
Let K be a compact Hausdorff space. Then (C( K ) δ ) ∗ is isometric-ally lattice isomorphic to M(Σ C K , N K ) . In particular, for every ϕ ∈ (C( K ) δ ) ∗ thereexists a unique µ ϕ ∈ M(Σ C K , N K ) so that ϕ (ˆ u ) = Z K udµ ϕ , ˆ u ∈ C( K ) δ , and the mapping S : (C( K ) δ ) ∗ ∋ ϕ µ ϕ ∈ M(Σ C K , N K ) is a linear lattice isometryonto M(Σ C K , N K ) .Proof. It follows immediately from [28, Theorem 2.3] that S is a linear isometry ontoM(Σ C K , N K ). That both S and its inverse are positive operators follows immediatelyfrom the construction of µ ϕ from ϕ ∈ (C( K ) δ ) ∗ and the definition of the integral.Indeed, for ϕ ∈ (C( K ) δ ) ∗ + and µ ∈ M(Σ C K , N K ) + µ ϕ ( B ) = ϕ (ˆ B ) , B ∈ Σ C K and Z K f dµ = sup s ∈ S f Z K sdµ, f ∈ B(Σ C K ) + where S f consists of all simple, positive functions dominated by f , see [18] and [28]for the details. (cid:3) Recall that Σ C K / N K is a complete Boolean algebra, isomorphic to R K . For B ∈ Σ C K let ˆ B denote the equivalence class in Σ C K / N K containing B . For B , B ∈ Σ C K ,ˆ B = ˆ B if and only if B ∆ B ∈ N K . Denote by M( R K ) the space of boundedfinitely additive measures on R K . This space is a Banach lattice with respect tothe pointwise order and variation norm, see for instance [12, 326Y (j)].Let ϕ ∈ M(Σ C K , N K ). Since ϕ [ N K ] = { } , ϕ induces a (signed) finitely additivemeasure µ ϕ on R K , µ ϕ ( ˆ B ) = ϕ ( B ) , B ∈ Σ C K . Conversely, every finitely additive measure µ on R K induces a measure ϕ µ ∈ M(Σ C K , N K ), ϕ µ ( B ) = µ ( ˆ B ) , B ∈ Σ C K . The maps M(Σ C K , N K ) ∋ ϕ µ ϕ ∈ M( R K ) and M( R K ) ∋ µ ϕ µ ∈ M(Σ C K , N K )are positive, linear isometries, and each is the inverse of the other. Therefore wehave the following. Corollary 4.3.
Let K be a compact Hausdorff space. The following statements aretrue. (i) The following spaces are pairwise isometrically lattice isomorphic: (C( K ) δ ) ∗ , M( R K ) , M(Σ C K , N K ) and M( G K ) . JAN HARM VAN DER WALT (iv) If K is Stonean then C( K ) ∗ , M( R K ) and M( K ) pairwise isometricallylattice isomorphic. All this is well (but perhaps not widely) known, see for instance [12, Chapter32]. That M(Σ C K , N K ) and M( G K ) are isometrically lattice isomorphic can also bederived from [28, Paragraph 4.5], and [12, 362A] may be used to show that M( R K )is isometrically lattice isomorphic to M( G K ).For the remainder of the paper we will be primarily concerned with the repres-entation of (C( K ) δ ) ∗ given in Theorem 4.2.5. Order continuous elements of (C( K ) δ ) ∗ and hyper-Stonean spaces The first result of this section is a characterisation of σ -order continuous func-tionals on C( K ) δ . In contrast with the classical representation theorem for function-als on C( K ) in terms of Borel measures, there is an exact correspondence betweencountably additivity of a measure and σ -order continuity of the corresponding func-tional. Theorem 5.1.
A measure ϕ ∈ (C( K ) δ ) ∗ is σ -order continuous if and only if it iscountably additive on Σ C K .Proof. Both the countably additive elements of M(Σ C K , N K ) and the σ -order con-tinuous elements in (C( K ) δ ) ∗ are ideals in the respective ambient spaces. Thereforewe may assume that ϕ ≥ ϕ is σ -order continuous. Let ( C n ) be a decreasing (with respect toinclusion) sequence of sets in Σ C so that T { C n : n ∈ N } = ∅ . For each n ∈ N let f n = C n . Then ˆ f n ↓ ˆ in L ∞ (Σ C K , N K ). Therefore ϕ ( C n ) = Z K ˆ f n dϕ −→ ϕ is countably additive.Conversely, assume that ϕ is countably additive. Consider a sequence ˆ f n ↓ ˆ inL ∞ (Σ C K , N K ). Replacing each f n with ( f ∧ . . . ∧ f n ) ∨ if necessary and using(3.1) we may assume that the sequence ( f n ) in B(Σ C K ) is pointwise decreasing andbounded below by 0. Let f : K ∋ x inf n ∈ N f n ( x ) ∈ R . Then f is Σ C K -measurableand bounded on K ; that is, f ∈ B(Σ C K ). But ˆ ≤ ˆ f ≤ ˆ f n for all n ∈ N . Thereforeˆ f = ˆ so that f − [ R + ] ∈ N . By the Lebesgue Dominated Convergence Theorem, ϕ ( ˆ f n ) = Z K f n dϕ −→ Z K f dϕ = 0 . so that ϕ is σ -order continuous. (cid:3) For a measure ϕ ∈ M(Σ C K , N K ) + we define regularity in the same way as forBorel measures. That is, ϕ is regular if for every B ∈ Σ B K ,sup { µ ( C ) : C ⊆ B is compact } = µ ( B ) = inf { µ ( U ) : U ⊇ B is open } . We note that each of the identities above implies the other. A general measure ϕ ∈ M(Σ C K , N K ) is regular if | ϕ | is regular. Theorem 5.2.
A measure ϕ ∈ (C( K ) δ ) ∗ is order continuous if and only if it iscountably additive and regular on Σ C K . ATEGORY MEASURES, THE DUAL OF C( K ) δ AND HYPER-STONEAN SPACES 9
Proof.
Let ϕ ≥
0. Assume that ϕ is order continuous. By Theorem 5.1, ϕ iscountably additive. Let A ∈ Σ C K . Then ˆ A ∈ L ∞ (Σ C K , N K ). Fix ǫ > D ǫ = { ˆ f ∈ L ∞ (Σ C K , N K ) : f ∈ C( K ) , (1 + ǫ/ A ≤ f } . According to Theorem3.5, D ǫ ↓ (1 + ǫ/ A in L ∞ (Σ C K , N K ). By order continuity of ϕ ,(1 + ǫ/ ϕ ( A ) = Z (1 + ǫ/ A dφ = inf ˆ f ∈ D ǫ Z K f dϕ. Therefore there exists ˆ f ∈ D ǫ so that Z K f dϕ < (1 + ǫ ) ϕ ( A ) . Let U = f − [(1 , ∞ )]. Then U ⊇ A is open and ϕ ( U ) = Z K U dϕ ≤ Z K f dϕ < (1 + ǫ ) ϕ ( A ) . Therefore ϕ ( A ) ≤ inf { ϕ ( U ) : U ⊇ A open } < (1 + ǫ ) ϕ ( A ). This holds for all ǫ > ϕ ( A ) = inf { ϕ ( U ) : U ⊇ A open } . Therefore ϕ is outer regular, henceregular.Conversely, assume that ϕ is countably additive and regular on Σ C K . Then therestriction ϕ of ϕ to Σ B K is countably additive and regular, and ϕ ( N ) = 0 forevery meagre Borel set N . Therefore ϕ is a normal Borel measure on K so thatthe restriction of the functional ϕ to C( K ) is order continuous. By Lemma 2.1 (i), ϕ is order continuous on C( K ) δ . (cid:3) Proposition 5.3.
Let ϕ ∈ (C( K ) δ ) ∗ + . Then ϕ is strictly positive if and only if, forevery every A ∈ Σ C K , ϕ ( A ) = 0 if and only if A ∈ N K .Proof. Assume that ϕ is strictly positive. Let A ∈ Σ C K \ N K . Then ˆ < ˆ A so that ϕ ( A ) = ϕ (ˆ A ) > A ∈ Σ C K , ϕ ( A ) = 0 if and only if A ∈ N K . Considerany ˆ < ˆ f ∈ L ∞ (Σ C K , N K ). Then there exists an ǫ > A = f − [[ ǫ, ∞ ]] ∈ Σ C K \ N K . Consequently, ϕ ( ˆ f ) ≥ Z K A dϕ = ϕ ( A ) > . Therefore ϕ is strictly positive. (cid:3) Theorems 2.5, 5.1 and 5.2, and Proposition 5.3 now yields the following result,establishing the relationship between order continuous functionals and categorymeasures.
Corollary 5.4.
Let K be a compact Hausdorff space. Then the following statementsare equivalent. (i) K satisfies property (***). (ii) K admits a category measure. (iii) C( K ) δ admits a strictly positive σ -order continuous linear functional. (iv) C( K ) admits a strictly positive σ -order continuous linear functional. (v) C( K ) admits a strictly positive order continuous linear functional. (vi) C( K ) δ admits a strictly positive order continuous linear functional. Proof.
That (i) and (ii) are equivalent is Theorem 2.5. The equivalence of (ii) and(iii) follows immediately from the definition of a category measure, Theorem 5.1and Proposition 5.3. That (iii) implies (iv) follows from Lemma 2.1 (ii).To see that (iv) implies (v), assume that C( K ) admits a strictly positive σ -ordercontinuous linear functional ϕ . Then there exists a fully supported regular Borelmeasure on K , hence K satisfies cc, see for instance [7, Proposition 4.1.6]. ThenC( K ) is order separable so that ϕ is order continuous.The equivalence of (v) and (vi) follows from Lemma 2.1 (i) and (ii). Lastly, that(vi) implies (iii) is obvious, which complete the proof. (cid:3) Remark . For the equivalence of (iii) to (vi) in Corollary 5.4 the assumption ofstrict positivity is essential. Indeed, there exists a compact Hausdorff space K anda σ -order continuous functional on C( K ) which is not order continuous [7, Example4.7.16]. Furthermore, the statement ‘there exists a Stonean space K and a σ -ordercontinuous functional on C( K ) which is not order continuous’ is equivalent to theexistence of a measurable cardinal, see [17].As an application of Corollary 5.4 we obtain the following topological character-ization of hyper-Stonean spaces. Theorem 5.6.
Let K be a Stonean space. Then K is hyper-Stonean if and onlyif there exists a collection U of clopen subsets of K so that S U is dense in K andevery U ∈ U satisfies property (***).Proof. Assume that K is hyper-Stonean. For each normal measure µ on K , let S µ denote the support of µ . Let U = { S µ : µ ∈ N( K ) } . By definition of a hyper-Stonean space S U is dense in K . Each S µ ∈ U is clopen, hence itself Stonean, and µ defines a strictly positive order continuous functional on C( S µ ). Corollary 5.4implies that each S µ satisfies property (***).Assume that there exists a collection U of clopen subsets of K so that S U isdense in K and every U ∈ U satisfies property (***). We claim that each U ∈ U isthe support of a normal measure on K . Fix U ∈ U . By Corollary 5.4 there existsa strictly positive order continuous functional ϕ on C( U ). Consider the linearfunctional ψ : C( K ) ∋ f ψ ( f | U ) ∈ R . Because U is clopen, hence regular closedin K , ψ is order continuous by [15, Theorem 3.4]. Therefore there exists a uniquenormal measure µ on K so that ψ ( u ) = Z K f dµ, f ∈ C( K ) . For every f ∈ C( K ) + , f | S µ = if and only if ψ ( f ) = 0, if and only if f | U = .Therefore U = S µ , which verifies our claim.Since S U is dense in K , the union of the supports of the normal measures on K is dense in K ; that is, K is hyper-Stonean. (cid:3) Remark . It is possible to obtain our main results, namely Theorem 4.2, fromwhich the results in Section 5 follow in a less direct manner. We briefly recall howthis may be achieved.In [8], de Jonge and van Rooij give a construction of C( K ) δ in terms of Borelmeasurable functions which is very similar to that given in Theorem 3.5. If Bdenotes the vector lattice of bounded Borel measurable functions on K and N thesubspace of B consisting of these functions which vanish of a meagre Borel set, thenC( K ) δ can be identified with D( K ) = B / N. Dales et al. [7] call the space D( K ) ATEGORY MEASURES, THE DUAL OF C( K ) δ AND HYPER-STONEAN SPACES 11 the Dixmier algebra of K . Using this construction our results can be obtained viathe machinery of measures on Boolean algebras as set out, for instance, in [12]. Wehave opted for a direct and more transparent approach. References
1. C. D. Aliprantis and O. Burkinshaw,
Locally solid Riesz spaces , Academic Press, New York-London, 1978.2. C.D. Aliprantis and O. Burkinshaw,
Positive operators , Springer, Dordrecht, 2006, reprint ofthe 1985 original.3. R. Anguelov,
Dedekind order completion of C ( X ) by Hausdorff continuous functions , Quaest.Math. (2004), no. 2, 153–169.4. S. A. Argyros, On compact spaces without strictly positive measure , Pacific J. Math. (1983), no. 2, 257–272.5. E. Behrends, R. Danckwerts, R. Evans, S. G¨obel, P. Greim, K. Meyfarth, and W. M¨uller, L p -structure in real Banach spaces , Lecture Notes in Mathematics, Vol. 613, Springer-Verlag,Berlin-New York, 1977.6. M. Cambern and p. Greim, Uniqueness of preduals for spaces of continuous vector functions ,Canad. Math. Bull. (1989), no. 1, 98–104.7. H. G. Dales, F. K. Dashiell, Jr., T.-M. Lau, and D. Strauss, Banach spaces of continuousfunctions as dual spaces , CMS Books in Mathematics/Ouvrages de Math´ematiques de laSMC, Springer, Cham, 2016.8. E. de Jonge and A. C. M. van Rooij,
Introduction to Riesz spaces , Mathematisch Centrum,Amsterdam, 1977.9. B. de Pagter and C. B. Huijsmans, On z -ideals and d -ideals in Riesz spaces. II , Nederl. Akad.Wetensch. Indag. Math. (1980), no. 4, 391–408.10. R. P. Dilworth, The normal completion of the lattice of continuous functions , Trans. Amer.Math. Soc. (1950), 427–438.11. J. Dixmier, Sur certains espaces consid´er´es par M. H. Stone , Summa Brasil. Math. (1951),151–182.12. D. H. Fremlin, Measure theory. volume 3 , Torres Fremlin, Colchester, 2002.13. A. M. Gleason,
Projective topological spaces , Illinois J. Math. (1958), 482–489.14. A. Horn, The normal completion of a subset of a complete lattice and lattices of continuousfunctions , Pacific J. Math. (1953), 137–152.15. M. Kandi´c and A. Vavpetiˇc, Topological aspects of order in C ( X ), Positivity (2019), no. 3,617–635.16. J. L. Kelley, Measures on Boolean algebras , Pacific J. Math. (1959), 1165–1177.17. W. A. J. Luxemburg, Is every integral normal? , Bull. Amer. Math. Soc. (1967), 685–688.18. , Integration with respect to finitely additive measures , Positive operators, Riesz spaces,and economics (Pasadena, CA, 1990), Springer, Berlin, 1991, pp. 109–150.19. W. A. J. Luxemburg and A. C. Zaanen,
Riesz spaces. Vol. I , North-Holland Publishing Co.,Amsterdam-London; American Elsevier Publishing Co., New York, 1971.20. F. Maeda and T. Ogasawara,
Representation of vector lattices , J. Sci. Hirosima Univ. Ser. A. (1942), 17–35.21. D. Maharam, Category, Boolean algebras and measure , Proceedings of the Fourth Prague To-pological Symposium, held in Prague, August 1976 (Josef Nov´ak, ed.), Society of CzechoslovakMathematicians and Physicists, Prague, 1977, Part B: Contributed papers, pp. 124–135.22. W. Maxey,
The dedekind completion of c ( x ) and its second dual , Ph.D. thesis, Purdue Uni-versity, 1973.23. J. C. Oxtoby, Spaces that admit a category measure , J. Reine Angew. Math. (1960/1961),156–170.24. ,
Measure and category , second ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980.25. H. H. Schaefer,
Banach lattices and positive operators , Springer-Verlag, New York-Heidelberg,1974, Die Grundlehren der mathematischen Wissenschaften, Band 215.26. J. M. Ayerbe Toledano,
Category measures on Baire spaces , Publ. Mat. (1990), no. 2,299–305.
27. J. H. van der Walt,
Representations of the Dedekind completions of spaces of continuous func-tions , Positivity and noncommutative analysis, Trends Math., Birkh¨auser/Springer, Cham,[2019] © Finitely additive measures , Trans. Amer. Math. Soc. (1952),46–66.(1952),46–66.