Probabilistic Powerdomains and Quasi-Continuous Domains
PProbabilistic Powerdomains andQuasi-Continuous Domains
Jean Goubault-Larrecq †† Université Paris-Saclay, ENS Paris-Saclay, CNRS, LSV, 91190,Gif-sur-Yvette, France. [email protected]
July 20, 2020
Abstract
The probabilistic powerdomain V X on a space X is the space of allcontinuous valuations on X . We show that, for every quasi-continuousdomain X , V X is again a quasi-continuous domain, and that the Scottand weak topologies then agree on V X . This also applies to the subspacesof probability and subprobability valuations on X . We also show that theScott and weak topologies on the V X may differ when X is not quasi-continuous, and we give a simple, compact Hausdorff counterexample. Continuous valuations are an alternative to measures, which are popular incomputer science, and notably in the semantics of programming languages [13,12]. The space of all continuous valuations on a topological space X is calledthe probabilistic powerdomain V X on X . It is known that the probabilisticpowerdomain of a directed-complete partial order (dcpo) is a dcpo again, inshort, V preserves dcpos; similarly, V preserves continuous dcpos, but failsto preserve complete lattices and bc-domains. All that was proved by Jones[13, 12]. It is unknown whether V preserves RB-domains or FS-domains, exceptin special cases [14]. On the positive side, V preserves stably compact spaces[14, 3], QRB-domains [8, 10], and coherent quasi-continuous dcpos [19]. (Thelatter two results are equivalent, since QRB-domains coincide with coherentquasi-continuous dcpos [17, 10], and also with Li and Xu’s QFS-domains [18].)Lyu and Kou [19] asked whether coherence was required, in other words,whether V preserves quasi-continuous, not necessarily coherent, dcpos. We showthat this is indeed the case, and that, in this case, the Scott and weak topologiesagree on the probabilistic powerdomain. We show this in Section 5, after a fewpreliminaries: general preliminaries in Section 2, some required material due toHeckmann on so-called point-continuous valuations in Section 3, and a useful1 a r X i v : . [ m a t h . GN ] J u l emma on capacities in Section 4. We refine the result and we handle the caseof probability continuous valuations in Section 6.Since every continuous dcpo is quasi-continuous, the coincidence of the Scottand weak topologies on V X , where X is quasi-continuous, generalizes a resultof Kirch [16, Satz 8.6], see also [20, Satz 4.10], according to which the Scott andweak topologies on V X agree for every continuous dcpo X . Alvarez-Manilla,Jung and Keimel asked whether they agree on V ≤ X for every stably compactspace X [3, Section 5, second open problem]. We will show that this is not thecase, through a simple, compact Hausdorff example in Section 7. Hence thesituation with quasi-continuous domains is probably rather exceptional. We refer to [7, 9] on domain theory and point-set topology, specially non-Hausdorff topology. Compactness does not involve separation.A dcpo (directed-complete partial order) is a poset P in which every directedfamily D has a supremum sup D . A Scott-open subset of P is a subset U thatis upwards-closed (for every x ∈ U and every y such that x ≤ y , y is in U ) andis such that, for every directed family D in P , if sup D ∈ U then D intersects U . The Scott-open subsets of P form a topology called the Scott topology .Every complete lattice is a dcpo. For example, R + def = R + ∪ {∞} , with theusual ordering that places ∞ above all non-negative real numbers, is a dcpo.The family of open subsets O X of a topological space is a dcpo, too.A Scott-continuous map f : P → Q between dcpos is a monotonic map thatpreserves suprema of directed families. A map from P to Q is Scott-continuousif and only if it is continuous with respect to the Scott topologies on P and Q .A valuation ν on a topological space X is a strict, modular, monotonic mapfrom O X to R + . That ν is strict means that ν ( ∅ ) = 0 . That it is modular means that ν ( U ) + ν ( V ) = ν ( U ∪ V ) + ν ( U ∩ V ) for all open subsets U and V .A continuous valuation is a valuation that is Scott-continuous from O X to R + .Continuous valuations and measures are close cousins. Every τ -smooth Borelmeasure defines a continuous valuation by restricting it to O X ; and every Borelmeasure on a hereditary Lindelöf space is τ -smooth [1]. Conversely, every con-tinuous valuation on an LCS-complete space extends to a measure on the Borel σ -algebra [5, Theorem 1.1]—an LCS-complete space is any subspace obtainedas a G δ subset of a locally compact sober space.We write V X for the dcpo of all continuous valuations on X , ordered by the stochastic ordering : µ ≤ ν if and only if µ ( U ) ≤ ν ( U ) for every U ∈ O X . V ≤ X (resp., V X ) is the subdcpo of all subprobability (resp., probability ) continuousvaluations ν , namely those such that ν ( X ) ≤ (resp., ν ( X ) = 1 ). We willusually write V ∗ X to denote any of those dcpos, where ∗ stands for nothing,“ ≤ ”, or “ ”.The weak topology on V ∗ X is the coarsest one that makes [ r (cid:28) U ] ∗ def = { ν ∈ V ∗ X | r (cid:28) ν ( U ) } open for every r ∈ R + and every U ∈ O X . Here (cid:28) is theso-called way-below relation on R + ; we have r (cid:28) s if and only if r = 0 or r < s .2he sets [ U > r ] ∗ with U ∈ O X and r ∈ R + (cid:114) { } form another subbase of theweak topology, since [ U > r ] ∗ = [ r (cid:28) U ] ∗ if r (cid:54) = 0 , and [0 (cid:28) U ] ∗ = V ∗ X . Wewrite V ∗ , w X for V ∗ X with the weak topology. The weak topology is coarserthan the Scott topology of the stochastic ordering ≤ .Every topological space X has a specialization preordering ≤ , defined by x ≤ y if and only if every open neighborhood of x contains y . A T space isone such that ≤ is an ordering. As examples, for every dcpo P , ordered by ≤ , the specialization preordering of P with its Scott topology is ≤ ; and thespecialization preordering of V ∗ X is the stochastic ordering.For every point x ∈ X , the closure of { x } coincides with the downwardclosure ↓ x def = { y ∈ X | y ≤ x } of x in the specialization preordering. In general,we write ↓ A for the downward closure of a set A , so that ↓ x = ↓{ x } .A subset A of a space X is saturated if and only if it is equal to the intersec-tion of its open neighborhoods, equivalently if it is upwards-closed with respectto the specialization preordering ≤ . We write ↑ A for the upward closure of A .For every compact subset K of X , ↑ K is compact saturated. This is thecase in particular if K is finite: we call the sets of the form ↑ E , with E finite, finitary compact . A space X is locally finitary compact if and only if it has abase consisting of interiors int ( ↑ E ) of finitary compact sets.The standard definition of a quasi-continuous dcpo is through the notion ofa so-called way-below relation between finite subsets. We will instead use thefollowing characterization [9, Exercise 8.3.39]: the quasi-continuous dcpos areexactly the locally finitary compact, sober spaces. Notably, every locally finitarycompact, sober space is a quasi-continuous dcpo in its specialization ordering ≤ ; also, the topology is exactly the Scott topology of ≤ .We have mentioned sober spaces a few times already. A closed subset C ofa space X is irreducible if and only if it is non-empty and, for all closed subsets C and C of X , if C ⊆ C ∪ C then C ⊆ C or C ⊆ C . The closures ↓ x ofpoints are always irreducible closed. A sober space is any T space in which theonly irreducible closed subsets are closures of points. R + is sober in its Scotttopology. Every quasi-continuous dcpo is sober in its Scott topology (by ourdefinition), every Hausdorff space is sober; also, V w X is sober for every space X [11, Proposition 5.1].The sober subspaces Y of a sober space X are exactly those that are closed inthe so-called Skula , or strong topology on X [15, Corollary 3.5]. That topology isthe coarsest one that contains both the original open and the original closed setsas open sets. We note that V ≤ , w X is closed in V w X , being the complementof [ X > . Every closed set is Skula-closed, so V ≤ , w X is also a sober space.Also, V , w X is the intersection of the closed set V ≤ , w X with the open sets [ X > − (cid:15) ] , (cid:15) > , hence is also Skula-closed and therefore sober as well.The forgetful functor from the category of sober spaces and continuous mapsto the category of topological spaces has a left adjoint called sobrification . Ex-plicitly, this means that every topological space X has a sobrification X s , whichis a sober topological space; there is a continuous map η X : X → X s , called the unit ; and every continuous map f : X → Y where Y is sober extends to a unique3ontinuous map ˆ f : X s → Y , in the sense that ˆ f ◦ η X = f . Concretely, X s canbe realized as the space of all irreducible closed subsets of X , with a suitabletopology, and η X ( x ) def = ↓ x . By Proposition 3.4 of [15], given any subspace Y of a sober space X , the Skula-closure cl s ( Y ) of Y in X is also a sobrification of Y , with η Y defined as the inclusion map. In general, for a T space Y , and asober space X , together with a continuous map f : Y → X , X is a sobrificationof Y with unit f if and only if f is a topological embedding, with Skula-denseimage [15, Proposition 3.2]. Among all the continuous valuations that exist on a space X , the simple valua-tions are those of the form (cid:80) x ∈ A a x δ x , where A is a finite subset of X , a x ∈ R + ,and δ x is the Dirac mass, defined by δ x ( U ) def = 1 if x ∈ U , otherwise. We let V ∗ , f X be the subspace of V ∗ , w X that consists of its simple valuations.Heckmann characterized the sobrification of V f X as being the space V p X of so-called point-continuous valuations on X [11, Theorem 5.5], together withinclusion as unit. Those are the valuations ν on X that are continuous from O p X to R + . O p X is the lattice of open subsets of X with the point topology,namely the coarsest topology that makes { U ∈ O X | x ∈ U } open for everypoint x ∈ X . We write V ∗ , p X for the usual variants. Lemma 3.1.
Let X be a topological space. V f X is Skula-dense in V p X .Proof. By Proposition 3.2 of [15], cited earlier: since V p X is a sobrification of V f X , with unit given by the inclusion map i , the image of i must be Skula-dense. Lemma 3.2.
Let X be a topological space and U be an open subset of V p X .For every ν ∈ U , there is a simple valuation ν (cid:48) in U such that ν (cid:48) ≤ ν .Proof. U ∩ ↓ ν is open in the Skula topology of V w X , and is non-empty, sinceit contains ν . Using Lemma 3.1, it must contain an element ν (cid:48) of V f X .Heckmann also showed that, when X is locally finitary compact, all contin-uous valuations are point-continuous, hence V w X = V p X [11, Theorem 4.1].Using that information, we obtain the following. Lemma 3.3.
Let X be a locally finitary compact space, U be an open subsetof V ∗ , w X , where ∗ is nothing or “ ≤ ”. For every ν ∈ U , there is a simplevaluation ν (cid:48) in U such that ν (cid:48) ≤ ν .Proof. When ∗ is nothing, this is Lemma 3.2, together with the fact that V w X = V p X .When ∗ is “ ≤ ”, we use the definition of the weak topology: ν is in some finiteintersection (cid:84) mi =1 [ U i > r i ] ≤ of subbasic open sets included in U . Then ν is alsoin the corresponding finite intersection (cid:84) mi =1 [ U i > r i ] of subbasic open sets of4 w X . We have just seen that there is a simple valuation ν (cid:48) ≤ ν in (cid:84) mi =1 [ U i > r i ] .Since ν (cid:48) ≤ ν , ν (cid:48) is a subprobability valuation, so ν (cid:48) is in (cid:84) mi =1 [ U i > r i ] ≤ , hencein U . Capacities are a generalization of valuations (or measures) introduced by Cho-quet [4], where modularity is abandoned in favor of weaker properties. We willneed the following kind.Given a subset B of a topological space, the unanimity game u B : O X → R + maps every open set U to if B ⊆ U , to otherwise. When B = { x } , u B issimply the Dirac mass δ x , but in general u B is not modular.We will consider functions κ of the form (cid:80) x ∈ A a x u B x , where A is a finitesubset of X , and for each x ∈ A , a x is a number in R + and B x is a finite non-empty subset of X , which we call simple capacities here. We compare capacities,and in general all functions from O X to R + , by κ ≤ ν if and only if κ ( U ) ≤ ν ( U ) for every U ∈ O X , extending the stochastic ordering from continuous valuationsto all maps.In this setting, an element f of Σ def = (cid:81) x ∈ A B x is a function that maps eachpoint x ∈ A to an element f ( x ) in B x . One can think of such functions f as strategies for picking an element of B x for each x ∈ A . We let ∆ Σ be the set of allfamilies (cid:126)β def = ( β f ) f ∈ Σ of non-negative real numbers such that (cid:80) f ∈ Σ β f = 1 . ∆ Σ is simply the standard n -simplex ∆ n def = { ( β , β , · · · , β n ) ∈ R n +1+ | (cid:80) ni =0 β i =1 } , where n is the cardinality of Σ minus .In order to show the following lemma, we will need to introduce the Choquetintegral (cid:82) x ∈ X h ( x ) dν of a lower semicontinuous map h : X → R + with respectto a set function ν : O X → R + . By definition, this is equal to the Riemannintegral (cid:82) ∞ ν ( h − (] t, ∞ ])) dt . Note that this makes sense, because h − (] t, ∞ ]) is open for every t ∈ R + , and because every non-increasing map is Riemann-integrable. In our setting, this form of the Choquet integral was introducedby Tix [20], and differs only slightly from Choquet’s original definition [4, Sec-tion 48]. Tix proved that, when ν is a continuous valuation, (cid:82) y ∈ X h ( y ) dν islinear and Scott-continuous in h [20, Lemma 4.2]. It is an easy exercise to ver-ify that (cid:82) y ∈ X χ U ( y ) dν = ν ( U ) for every open subset U of X , where χ U is thecharacteristic map of U . It follows that, when h is of the form (cid:80) mj =0 α j χ U j , (cid:82) y ∈ X h ( y ) dν = (cid:80) mj =0 α j ν ( U j ) .For a simple capacity κ def = (cid:80) x ∈ A a x u B x , we compute (cid:82) y ∈ X h ( y ) dκ as fol-lows. For each x ∈ A , (cid:82) y ∈ X h ( y ) d u B x = (cid:82) ∞ u B x ( h − (] t, ∞ ])) dt by the Choquetformula. But u B x ( h − (] t, ∞ ])) = 1 if and only if B x ⊆ h − (] t, ∞ ]) , if and onlyif min y ∈ B x h ( y ) > t . It follows that (cid:82) y ∈ X h ( y ) d u B x = min y ∈ B x h ( y ) . Hence (cid:82) y ∈ X h ( y ) dκ = (cid:80) x ∈ A a x min y ∈ B x h ( y ) . 5 emma 4.1. Let X be a topological space, and κ def = (cid:80) x ∈ A a x u B x be a simplecapacity on X .Let ν be any bounded continuous valuation on X . If κ ≤ ν , then, for some (cid:126)β ∈ ∆ Σ , (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) ≤ ν .Proof. This is a consequence of von Neumann’s original minimax theorem [21],which says that given any n × m matrix M with real entries, min (cid:126)α ∈ ∆ m max (cid:126)β ∈ ∆ n (cid:126)β (cid:124) M (cid:126)α = max (cid:126)β ∈ ∆ n min (cid:126)α ∈ ∆ m (cid:126)β (cid:124) M (cid:126)α. (1)In particular: ( † ) if for every (cid:126)α ∈ ∆ m , there is a (cid:126)β ∈ ∆ n such that (cid:126)β (cid:124) M (cid:126)α ≥ (namely, if the left-hand side of (1) is non-negative), then there is a (cid:126)β ∈ ∆ n such that, for every (cid:126)α ∈ ∆ m , (cid:126)β (cid:124) M (cid:126)α ≥ .We first show that: ( ∗ ) given finitely many open subsets U , U , . . . , U m of X ,we can find a (cid:126)β ∈ ∆ Σ such that, for every j , ≤ j ≤ m , (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) ( U j ) ≤ ν ( U j ) .Let κ denote (cid:80) x ∈ A a x u B x . Since κ ≤ ν , for every lower semicontinuous map h , (cid:82) y ∈ X h ( y ) dκ = (cid:82) ∞ κ ( h − (] t, ∞ ])) dt ≤ (cid:82) ∞ ν ( h − (] t, ∞ ])) dt = (cid:82) y ∈ X h ( y ) dν .In other words, (cid:80) x ∈ A a x min y ∈ B x h ( y ) ≤ (cid:82) y ∈ X h ( y ) dν .For every (cid:126)α ∈ ∆ m , we consider h (cid:126)α def = (cid:80) mj =0 α j χ U j for h . The inequality wehave just shown can be rewritten as (cid:80) x ∈ A a x min y ∈ B x h (cid:126)α ( y ) ≤ (cid:80) mj =0 α j ν ( U j ) .For each x ∈ A , there is an element y ∈ B x that makes h (cid:126)α ( y ) minimal, andwe call it f (cid:126)α ( x ) . Therefore (cid:80) x ∈ A a x h (cid:126)α ( f (cid:126)α ( x )) ≤ (cid:80) mj =0 α j ν ( U j ) . By defini-tion of h (cid:126)α , and since χ U j ( f (cid:126)α ( x )) = δ f (cid:126)α ( x ) ( U j ) , this can be written equiva-lently as (cid:80) x ∈ A (cid:80) mj =0 α j a x δ f (cid:126)α ( x ) ( U j ) ≤ (cid:80) mj =0 α j ν ( U j ) . It follows that thereis a vector (cid:126)β in ∆ Σ such that, for every j , ≤ j ≤ m , (cid:80) mj =0 α j ν ( U j ) − (cid:80) mj =0 α j (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) ( U j ) ≥ : namely, β f def = 1 if f = f (cid:126)α , and β f def = 0 otherwise.That can also be written as (cid:80) f ∈ Σ , ≤ j ≤ m α j β f (cid:0) ν ( U j ) − (cid:80) x ∈ A a x δ f ( x ) ( U j ) (cid:1) ≥ , hence as (cid:126)β (cid:124) M (cid:126)α ≥ for some matrix M . Using ( † ) , there is a vector (cid:126)β ∈ ∆ Σ such that, for every (cid:126)α ∈ ∆ m , (cid:126)β (cid:124) M (cid:126)α ≥ , in other words (cid:80) mj =0 α j ν ( U j ) − (cid:80) mj =0 α j (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) ( U j ) ≥ . In particular, for each j , ≤ j ≤ m ,taking (cid:126)α such that α j def = 1 and all its other components are , ν ( U j ) ≥ (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) ( U j ) . This proves ( ∗ ) .For every finite family A of open subsets of X , let C A be the set of vectors (cid:126)β ∈ ∆ Σ such that (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) ( U ) ≤ ν ( U ) for every U ∈ A . Claim ( ∗ ) above states that C A is non-empty (when A is non-empty; when A isempty, this is vacuously true). It is also a closed subset of ∆ Σ . The family ( C A ) A∈ P fin ( O X ) then has the finite intersection property: given any finite collec-tion of elements A , . . . , A k in P fin ( O X ) , (cid:84) ki =1 C A i = C (cid:83) ki =1 A i is non-empty.Since ∆ σ is compact, the intersection (cid:84) A∈ P fin ( O X ) C A is non-empty. Let (cid:126)β beany vector in that intersection. For every U ∈ O X , since (cid:126)β is in C { U } , we have (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) ( U ) ≤ ν ( U ) , and we conclude.6 The main theorem
We come to our main theorem. It applies in particular to every quasi-continuousdcpo X , namely to every locally finitary compact sober space, as we have an-nounced; but sobriety is not needed. We spend the rest of the section provingit. Theorem 5.1.
For every locally finitary compact space X , V w X = V p X and V ≤ , w X = V ≤ , p X are compact, locally finitary compact, sober spaces. Inparticular, they are quasi-continuous dcpos and the weak topology coincides withthe Scott topology.The sets of the form int ( ↑ E ) , where E ranges over the finite non-empty setsof simple (resp., simple subprobability) valuations form a base of the topology. Let ∗ be nothing or “ ≤ ”. We recall that the equality V w X = V p X holds for every locally finitary compact space X , as shown by Heckmann [11,Theorem 4.1]. The equality V ≤ , w X = V ≤ , p X immediately follows from it.We also recall that the quasi-continuous dcpos are exactly the locally finitarycompact sober spaces, and in particular that their topology must be the Scotttopology. The fact that V ∗ , w X is compact follows from the fact that it has aleast element in the stochastic ordering, namely the zero valuation: every opencover ( U i ) i ∈ I of V ∗ , w must be such that some U i contains the zero valuation, andtherefore coincide with the whole of V ∗ , w X , since open sets are upwards-closed.Finally, we recall that V ∗ , w X is sober.Therefore, it remains to show that V ∗ , w X is locally finitary compact. In therest of this section, we fix ν ∈ V ∗ , w X , and an open neighborhood U of ν in theweak topology. Then ν is in some finite intersection (cid:84) ni =1 [ U i > r i ] ∗ included in U , where each U i is open in X and r i ∈ R + (cid:114) { } . We will find a finite set E ofsimple valuations and an open subset V of V ∗ , w X such that ν ∈ V ⊆ ↑ E ⊆ U .Let us simplify the problem slightly. By Lemma 3.3, there is a simple valua-tion ν (cid:48) ≤ ν in (cid:84) ni =1 [ U i > r i ] ∗ . Hence, without loss of generality, we may assumethat ν itself is a simple valuation (cid:80) x ∈ A a x δ x , where A is a finite subset of X ,and a x ∈ R + (cid:114) { } for every x ∈ A .Since ν ( U i ) > r i for every i , ≤ i ≤ n , there is a number a ∈ ]0 , suchthat a.ν ( U i ) > r i for every i . There is also a positive number s i such that a.ν ( U i ) > s i > r i . We will need those numbers a and s i only near the end ofthe proof.Let us define a suitable open set V . For each point x ∈ A , let I x def = { i ∈ I | x ∈ U i } . Then (cid:84) i ∈ I x U i (cid:114) ↓ ( A (cid:114) ↑ x ) is an open neighborhood of x . It is easyto see that x is in (cid:84) i ∈ I x U i , but perhaps a bit less easy to see that x is not in ↓ ( A (cid:114) ↑ x ) : otherwise there woud be an element y ∈ A (cid:114) ↑ x above x , and thatis impossible.Since X is locally finitary compact, for each x ∈ A , one can find a finite set B x such that x ∈ int ( ↑ B x ) ⊆ ↑ B x ⊆ (cid:84) i ∈ I x U i (cid:114) ↓ ( A (cid:114) ↑ x ) . We will require abit more, and we will make sure that B x is also included in int ( ↑ B y ) for every y ∈ A such that y ≤ x . This can be done by finding B x in stages, starting7rom the lowest elements x of A and going up. Formally, since A is finite, wedefine B x by course-of-values induction on the number of elements y ∈ A suchthat y ≤ x , as follows: for each x ∈ A , we simply find a finite set B x suchthat x ∈ int ( ↑ B x ) ⊆ ↑ B x ⊆ (cid:84) i ∈ I x U i (cid:114) ↓ ( A (cid:114) ↑ x ) ∩ (cid:84) y ∈ A,y For all x, y ∈ A with x ≤ y , ↑ B y ⊆ V x ⊆ ↑ B x . Lemma 5.3. For every x ∈ A , for every i ∈ I , if x ∈ U i , then B x ⊆ U i .Proof. If x ∈ U i , then i ∈ I x . Since B x ⊆ (cid:84) i ∈ I x U i , the claim follows. Lemma 5.4. For all x, y ∈ A , x ∈ V y if and only if y ≤ x .Proof. If y ≤ x , and since V y is an open neighborhood of y , and is in particularupwards-closed, x is also in V y . If y (cid:54)≤ x , then x is in A (cid:114) ↑ y , hence in ↓ ( A (cid:114) ↑ y ) .It follows that x cannot be in (cid:84) i ∈ I y U i (cid:114) ↓ ( A (cid:114) ↑ y ) , hence cannot be in the smallerset V y . Definition 5.5 ( V ) . Let P ↑ A denote the (finite) family of upwards-closed sub-sets of A . For each B ∈ P ↑ A , let V B def = (cid:83) x ∈ B V x . Let also s B def = a. (cid:80) x ∈ B a x .The open set V is (cid:84) B ∈ P ↑ A [ s B (cid:28) V B ] . Recall that µ ∈ [ s B (cid:28) V B ] if and only if s B (cid:28) µ ( V B ) , if and only if s B = 0 or s B < µ ( V B ) . Lemma 5.6. ν ∈ V .Proof. For every B ∈ P ↑ A , we claim that A ∩ V B = B . For every x ∈ B , V x isincluded in V B , and since V x is an open neighborhood of x , it follows that x isin V B ; x is also in A , since B ⊆ A . Conversely, if x ∈ A ∩ V B , then x is in V y forsome y ∈ B . Both x and y are in A , so by Lemma 5.4, we obtain that y ≤ x .Since B is upwards-closed, x is in B .Let us verify that ν is in V , namely that, for every B ∈ P ↑ A , s B (cid:28) ν ( V B ) .Indeed, ν ( V B ) = (cid:80) x ∈ A ∩ V B a x = (cid:80) x ∈ B a x , since A ∩ V B = B . Now, since a < , a. (cid:80) x ∈ B a x (cid:28) (cid:80) x ∈ B a x . In other words, s B (cid:28) ν ( V B ) , as desired.Finding the finite set E is more difficult. As a first step in that direction,let κ def = a. (cid:80) x ∈ A a x u B x , and let us consider the set Q of all the continuousvaluation µ ∈ V ∗ X such that κ ≤ µ . Lemma 5.7. V ⊆ Q .Proof. Let µ be any element of V . We must show that, for every open subset U of X , a. (cid:80) x ∈ A,B x ⊆ U a x ≤ µ ( U ) .Let B def = { x ∈ A | B x ⊆ U } . For every x ∈ B and every y ∈ A with x ≤ y ,we have B y ⊆ ↑ B x ⊆ U by Lemma 5.2, so y is in B . Hence B is upwards-closedin A . 8hen the left-hand side a. (cid:80) x ∈ A,B x ⊆ U a x is just s B . Since µ ∈ V , s B (cid:28) µ ( V B ) . We recall that V B = (cid:83) x ∈ B V x , that V x is included in ↑ B x for each x , and that (by the definition of B ), ↑ B x is included in U for every x ∈ B .Therefore V B ⊆ U , and hence µ ( V B ) ≤ µ ( U ) , which concludes the proof.Let Σ def = (cid:81) x ∈ A B x , and ∆ Σ be the associated standard simplex. Lemma 5.7,together with Lemma 4.1, immediately implies the following. Lemma 5.8. Every element µ of V is above a simple valuation of the form a. (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) , for some (cid:126)β ∈ ∆ Σ . Let E be the set of simple valuations obtained this way, namely the set ofsimple valuations a. (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) , where (cid:126)β ∈ ∆ Σ . We have just shownthat every element µ of V is above some element of E .Note that the elements (cid:36) of E can all be written as (cid:80) z ∈ Z c z δ z , where Z def = (cid:83) x ∈ A B x , and c z ∈ R + . For each such (cid:36) , let (cid:36) be (cid:80) z ∈ Z N (cid:98) N c z (cid:99) , where N is a fixed, large enough (in particular, non-zero) natural number that we willdetermine shortly. Definition 5.9 ( E ) . E is the set of all simple valuations (cid:36) , where (cid:36) rangesover E . Lemma 5.10. E is a finite set.Proof. Z is finite and the coefficients N (cid:98) N c z (cid:99) are integer multiples of N between and (cid:80) x ∈ A a x . Lemma 5.11. V ⊆ ↑ E .Proof. For every z ∈ Z , and every c z ∈ R + , N (cid:98) N c z (cid:99) ≤ c z . It follows that (cid:36) ≤ (cid:36) for every (cid:36) ∈ E . Since every element of V is above some element (cid:36) of E by Lemma 5.8, it is also above the corresponding element (cid:36) of E . Lemma 5.12. ↑ E ⊆ U .Proof. We show that E is included in (cid:84) ni =1 [ U i > r i ] ∗ . For every x ∈ A , for every y ∈ B x , we have δ y ≥ u B x , simply because every open neighborhood of B x mustcontain x . Hence, for every (cid:36) ∈ E , say (cid:36) = a. (cid:80) f ∈ Σ ,x ∈ A β f a x δ f ( x ) , where (cid:126)β ∈ ∆ Σ , we have (cid:36) ≥ a. (cid:80) f ∈ Σ ,x ∈ A β f a x u B x = a. (cid:80) x ∈ A ( (cid:80) f ∈ Σ β f ) a x u B x = a. (cid:80) x ∈ A a x u B x = κ . For every i , ≤ i ≤ n , Lemma 5.3 states that for every x ∈ A , if x ∈ U i then B x is included in U i . Therefore (cid:36) ( U i ) = a. (cid:80) x ∈ A,B x ⊆ U i a x ≥ a. (cid:80) x ∈ A ∩ U i a x = a.ν ( U i ) . We now remember that a.ν ( U i ) > s i > r i . Inparticular, (cid:36) ( U i ) > s i .It is time we fixed the value of N . The values of c z and of N (cid:98) N c z (cid:99) differ by N at most, so for any open set U , the values (cid:36) ( U ) and (cid:36) ( U ) differ by N | Z | atmost, where | Z | is the cardinality of Z . It follows that (cid:36) ( U i ) > s i − N | Z | . Bypicking any non-zero natural number N larger than or equal to | Z | s i − r i for every i , ≤ i ≤ n , we therefore ensure that (cid:36) ( U i ) > r i for every i , hence that (cid:36) is in U . Since that holds for every (cid:36) ∈ E , E is included in U , hence also ↑ E .9ence, as promised, ν ∈ V (Lemma 5.6) ⊆ ↑ E (Lemma 5.11) ⊆ U (Lemma 5.12),where V is open (Definition 5.5) and E is finite (Lemma 5.10). This concludesthe proof of Theorem 5.1. We now apply the previous results to the space V , w X of probability continuousvaluations. A space X is pointed if and only if it has a least element ⊥ in itsspecialization preordering. We are not assuming X to be T , so ↓ ⊥ is a closedsubset that may be different from {⊥} . The open subsets of X (cid:114) ↓ ⊥ are justthe proper open subsets of X .The following is Edalat’s lifting trick , which was introduced in [6, Section 3]for dcpos, and in [2, Section 7.4] for stably locally compact spaces. Everycontinuous valuation ν on X gives rise to a continuous valuation ν − on X (cid:114) ↓ ⊥ by ν − ( U ) def = ν ( U ) for every U ∈ O ( X (cid:114) ↓ ⊥ ) . If ν ∈ V X , then ν − is in V ≤ X ,and we have much more, as we now show. Lemma 6.1. Let X be a pointed topological space, with least element ⊥ . Themap ν (cid:55)→ ν − is a homeomorphism of V , w X onto V ≤ , w ( X (cid:114) ↓ ⊥ ) . Its inversemaps every subprobability continuous valuation µ on X (cid:114) ↓ ⊥ to µ + , defined by µ + ( U ) def = µ ( U (cid:114) ↓ ⊥ ) + (1 − µ ( X (cid:114) ↓ ⊥ )) δ ⊥ , for every U ∈ O ( X (cid:114) ↓ ⊥ ) .Proof. Let ν ∈ V X . For every U ∈ O X , ( ν − ) + ( U ) = ν − ( U (cid:114) ↓ ⊥ ) + (1 − ν − ( X (cid:114) ↓ ⊥ )) δ ⊥ ( U ) . If U is a proper open subset of X , then U does not contain ⊥ , so U (cid:114) ↓ ⊥ = U , and δ ⊥ ( U ) = 0 , so ( ν − ) + ( U ) = ν − ( U ) = ν ( U ) . If U = X ,then ( ν − ) + ( U ) = ν − ( X (cid:114) ↓ ⊥ ) + (1 − ν − ( X (cid:114) ↓ ⊥ )) = 1 , and this is equal to ν ( U ) since U = X and ν ∈ V X .For every U ∈ O ( X (cid:114) ↓ ⊥ ) , ( µ + ) − ( U ) = µ + ( U ) = µ ( U ) , since U (cid:114) ↓ ⊥ = U ,and ⊥ is not in U .Hence the two maps ν (cid:55)→ ν − and µ (cid:55)→ µ + are inverse of each other.For every open subset U of X and every r ∈ R + (cid:114) { } , the inverse image of [ U > r ] by µ (cid:55)→ µ + is equal to one of the following sets. If U = X and r < ,this is the whole of V ≤ , w X . If U = X and r ≥ , this is empty. Finally, if U is a proper subset of X , hence does not contain ⊥ , then this is the set of all µ ∈ V ≤ ( X (cid:114) ↓ ⊥ ) such that µ + ( U ) > r , where µ + ( U ) = µ ( U ) : hence this is [ U > r ] ≤ . In any case, that inverse image is open, so µ (cid:55)→ µ + is continuous.For every open subset U of X (cid:114) ↓ ⊥ , for every r ∈ R + (cid:114) { } , the inverseimage of [ U > r ] ≤ by ν (cid:55)→ ν − is [ U > r ] . Therefore ν (cid:55)→ ν − is continuous.Lemma 6.1 allows us to obtain the following corollary to Theorem 5.1. Corollary 6.2. For every locally finitary compact, pointed space X , V , w X is compact, locally finitary compact, and sober. In particular, it is a quasi-continuous dcpo, and the weak topology coincides with the Scott topology.The sets of the form int ( ↑ E ) , where E ranges over the finite non-empty setsof simple probability valuations form a base of the topology. The Scott and weak topologies may differ The Scott and weak topologies on V ∗ X seem to agree in many situations, andAlvarez-Manilla, Jung and Keimel asked whether they agree on V ≤ X for everystably compact space X [3, Section 5, second open problem]. We show that thisis not the case.Let α ( N ) be the one-point compactification of the discrete space N . Itselements are the natural numbers, plus a fresh element ∞ . Its open subsets arethe subsets of N (not containing ∞ ), plus all the subsets α ( N ) (cid:114) E , where E ranges over the finite subsets of N . A discrete valuation on α ( N ) is any valuationof the form (cid:80) n ∈ N a n δ n + a ∞ δ ∞ , where each a n and a ∞ are in R + . They are allcontinuous valuations. Lemma 7.1. Letting ∗ be “ ≤ ” or “ ”.(i) The continuous valuations ν on α ( N ) are exactly the discrete valuations.(ii) The function f : V ∗ ( α ( N )) → Y ∗ that maps (cid:80) n ∈ N a n δ n + a ∞ δ ∞ to ( a x ) x ∈ α ( N ) is an order-isomorphism onto the poset Y ∗ of families of non-negative realnumbers whose sum is at most (if ∗ is “ ≤ ”) or exactly (if ∗ is “ ”),ordered pointwise.(iii) The set V of families ( a x ) x ∈ α ( N ) of Y ∗ such that a ∞ > is Scott-open in Y ∗ ,but f − ( V ) does not contain any basic open neighborhood (cid:84) ni =1 [ U i > r i ] ∗ of δ ∞ .Proof. (i) Let ν be any continuous valuation over α ( N ) . We recall that everycontinuous valuation on an LCS-complete space extends to a measure on theBorel σ -algebra [5, Theorem 1.1]. Every locally compact sober space is G δ initself, hence LCS-complete. Since every Hausdorff space is sober, and clearlylocally compact, α ( N ) is LCS-complete, and therefore ν extends to a measure ˜ ν on the Borel σ -algebra of α ( N ) . It is easy to see that the latter σ -algebrais the whole of P ( N ) . We define a n def = ˜ ν ( { n } ) = ν ( { n } ) , and a ∞ def = ˜ ν ( {∞} ) .By σ -additivity, for every (necessarily measurable) subset E of α ( N ) , ˜ ν ( E ) = (cid:80) x ∈ E a x . In particular, for every open subset U of α ( N ) , ν ( U ) = (cid:80) x ∈ U a x =( (cid:80) n ∈ N a n δ n + a ∞ δ ∞ )( U ) .(ii) Let ν be any element of V ∗ ( α ( N )) , and ˜ ν be a measure that extends ν tothe Borel σ -algebra. In a more precise way as in the statement of the lemma, wedefine f ( ν ) as ( a x ) x ∈ α ( N ) , as given in item (i), so that ν = (cid:80) n ∈ N a n δ n + a ∞ δ ∞ .This defines a bijection f of V ∗ ( α ( N )) onto Y ∗ .Since {∞} = (cid:84) n ∈ N V n , where V n is the open set { n, n + 1 , · · · , ∞} , and since ˜ ν is a bounded measure, a ∞ = ˜ ν ( {∞} ) = inf n ∈ N ˜ ν ( V n ) = inf n ∈ N ν ( V n ) . Thisimplies that a ∞ grows as ν grows. It is clear that a n = ν ( { n } ) grows, too, as ν grows. Therefore f is monotonic, and its inverse is clearly monotonic as well.(This discussion is superfluous when ∗ is “ ”, by the way, since in that case theordering on V ( α ( N )) and on Y is just equality.)(iii) V is clearly Scott-open in Y ∗ . We now imagine that f − ( V ) contains abasic open neighborhood (cid:84) ni =1 [ U i > r i ] ∗ of δ ∞ , where each U i is open in α ( N ) ,11nd r i ∈ R + (cid:114) { } . Since δ ∞ ∈ [ U i > r i ] ∗ , U i must contain ∞ (and r i < ),so U i = α ( N ) (cid:114) E i for some finite subset E i of N . Let n be a natural numberthat is not in any of the finite sets E i , ≤ i ≤ n . Then δ n ( U i ) = 1 > r i , so δ n is in (cid:84) ni =1 [ U i > r i ] ∗ , hence in f − ( V ) . However, f ( δ n ) is the family ( a x ) x ∈ α ( N ) such that a x = 0 for every x ∈ α ( N ) except for a n = 1 ; in particular, a ∞ = 0 ,showing that f ( δ n ) is not in V : contradiction. Theorem 7.2. Let ∗ be nothing, “ ≤ ” or “ ”. The Scott topology on V ∗ ( α ( N )) is strictly finer than the weak topology.Proof. We recall that the Scott topology on any space of the form V ∗ X is alwaysfiner than the weak topology.When ∗ is “ ≤ ” or “ ”, this is Lemma 7.1, item (iii): f − ( V ) is a Scott-openneighborhood of δ ∞ in V ∗ ( α ( N )) that is not open in the weak topology.When ∗ is nothing, we notice that V ≤ ( α ( N )) is Scott-closed in V ( α ( N )) .This easily implies that the Scott topology on V ≤ ( α ( N )) is the subspace topol-ogy induced by the Scott topology on V ( α ( N )) . If the latter agreed with theweak topology, then the Scott topology on V ≤ ( α ( N )) would be the subspacetopology induced by the inclusion in V w ( α ( N )) . But the latter is just the weaktopology on V ≤ ( α ( N )) , and we have just seen that it differs from the Scotttopology.The gap between the Scott and weak topologies on V ( α ( N )) is really enor-mous. By Corollary 37 of [3], V ≤ X and V X are stably compact for any stablycompact space X . This applies to X def = α ( N ) , since every compact Hausdorffspace is stably compact. One checks easily (e.g., by using Lemma 7.1, item (ii))that the stochastic ordering on V ( α ( N )) is simply equality, hence that the Scotttopology is the discrete topology. But the only discrete spaces that are (stably)compact are finite, and V ( α ( N )) is far from finite.The coincidence of the Scott and weak topologies of Theorem 5.1, and firstobserved by Kirch in the case where X is a continuous dcpo, is probably excep-tional. We leave open the question of the exact characterization of those spaces X for which the weak and Scott topologies agree on V ∗ X . Acknowledgments My deepest thanks to Xiaodong Jia, who found a mistake in a previous versionof Lemma 4.1, and another one in a previous version of Lemma 5.7. References [1] Wolfgang Adamski. τ -smooth Borel measures on topological spaces. 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