AN EXTENSION RESULT FOR CONTINUOUS VALUATIONS M. ALVAREZ-MANILLA, A. EDALAT AND N. SAHEB-DJAHROMI Abstract We show, by a simple and direct proof, that if a bounded valuation on a monotone convergence space is the supremum of a directed family of simple valuations then it has a unique extension to a Borel measure. In particular this holds for any directed complete partial order (dcpo) with the Scott topology. It follows that every bounded and continuous valuation on a continuous dcpo can be extended uniquely to a Borel measure. The last result also holds for - nite valuations, but fails for dcpo's in general.
1 Introduction The motivation of the present work goes back to Horn and Tarski [11] who gave conditions for extending a measure de ned on a subalgebra A of a given Boolean algebra L to the whole algebra. Pettis [21] continued this work for the case L is a lattice and is a valuation. The term valuation is borrowed from Birkho [4] to designate a nonnegative real function , de ned on a lattice L, such that it is modular, monotone and (0) = 0. Pettis gave the necessary conditions for extending a valuation to a measure on the -algebra generated by L. He showed applications of this result in the proof of the Riesz-Marko theorem and the existence of the Haar measure. In this paper a valuation is a real valued function de ned on the lattice of open sets of a given topological space that is modular, monotone and such that (;) = 0 (see below); a continuous valuation is one which preserves directed suprema. For a directed complete partial orders (dcpo), the problem of extending a continuous valuation to a Borel measure reappeared in the context of probabilistic nondeterminism [23]. In [23] a proof was given for !-algebraic dcpo's but it contained a gap. Norberg established the result for 1991 Mathematics Subject Classi cation 60B05, 06B35, 54F05
1
- nite valuations on !-continuous dcpo's and gave applications in random
set theory and a proof of the Daniell-Kolmogorov theorem for continuous lattices. Lawson [16] showed that a continuous valuation de ned on a distributive continuous lattice L has a unique extension to a regular Borel measure on L (on the Borel -algebra of its Lawson topology). It follows that every continuous valuation on a locally quasicompact sober topological space extends uniquely to a measure, provided that the space is second countable or the patch topology is compact. Norberg and Vervaat gave an extension theorem for modular capacities de ned on quasicompact sets [20, Theorem 3.7]. For coherent spaces such capacities and continuous valuations are equivalent notions ([20, Theorem 5.4] and [26, Chapter 3]). As an immediate consequence it follows that a valuation on a coherent space extends uniquely to a Borel measure (on the -algebra generated by the patch topology). This generalises Lawson's latter result. These results imply the extension result for ! -continuous dcpo's and coherent continuous dcpo's. Finally Jones and Plotkin ([14, Theorem 8.2] and [13, Chapter 5]), following the approach in [23], claimed the result for continuous dcpo's, without presenting a correct proof. Continuous valuations have in recent years played a key role in the domain theoretic approach to classical measure theory which has led to a new generalisation of the Riemann integral with applications in computation in fractal geometry [7]. In this paper we give a short and direct proof, based on the original idea in [23]. We show that if a - nite valuation on a monotone convergence space (see below), is the supremum of a directed family of simple valuations then it has a unique extension to a Borel measure. As a corollary we have that every bounded and continuous valuation on a continuous dcpo can be extended uniquely to a Borel measure. We also show that the extension fails in general for a continuous valuation on a dcpo.
2 Basic de nitions and auxiliary results We recall some basic de nitions, our main references for domain theory are [8, 12, 1]. A topological space is a pair (X; ) where X is a set and is a topology for X . We denote by B(X ) the Borel ?algebra of (X; ). A Borel measure is nite if (X ) < 1 and - nite if there exists a countable S family fCigi2N of sets in B(X ) such that X = i2N Ci and (Ci ) < 1 for all i 2 N. We say that a measure is normalised if (X ) = 1. Observe that 2
every nite measure with (X ) > 0 can be normalised. For the measurable space (X; B(X )) we denote by MX the set of all positive Borel measures bounded by 1 and by M1X the set of all probability measures. Let (P; v) be a partially ordered set (poset). For A P we de ne # A = fx 2 P j 9a 2 A:x v ag. We often abbreviate # fag by # a. In a similar way we de ne " A and " a. The set A is lower if A =# A and upper if A =" A. A nonempty subset A P is said to be directed if for all x; y 2 A there exists z 2 A such that x; y v z . A nonempty subset T P is called a chain if for all x; y 2 T we have x v y or y v x. A directed complete partial order is a poset P inFwhich every directed subset A has a least upper bound (lub) denoted by " A. The Scott topology of a poset (P; v) is de ned as follows: Q P is open if Q is upper and for all directed subsets F F A P such that " A exists, we have " A 2 Q implies that Q \ A 6= ;. Note that # a is closed for all a 2 P . The Scott topology will be denoted by
P . A dcpo with the Scott topology is a T0 space. Conversely, any T0 topological space (X; ) can be partially ordered via the specialisation order. Let cl(B ) denote the topological closure of any B X . The specialisation order is de ned by x v y if and only if x 2 cl(fyg). We de ne a monotone convergence space [8, Chapter II,3.9] as a triple (X; ; v) satisfying: (X; ) is a T0 space with v the specialisation order, (X; v) is a dcpo and is coarser than the Scott topology. It is easy to see that in a monotone convergence space every directed set A converges F to its supremum, i.e. for all open sets O such that " A 2 O there exists x 2 A such that for all y 2 A if x v y, then y 2 A. A dcpo (D; v) with the Scott topology is a monotone convergence space and the specialisation order coincides with v. Assume for the rest of this section that (X; ) is any topological space. A closed subset C of X is called irreducible, if for any closed subsets C1; C2 of X with C C1 [ C2, we have C C1 or C C2. We say X is sober if all closed irreducible subsets of X are the closure of a unique point. A sober space is always a monotone convergence space (hence T0 ), but there exist monotone convergence spaces that are not sober (see the last section of the paper). A subset Q of X is called quasicompact if it has the Heine-Borel property, i.e. every open cover of Q has a nite subcover. X is compact if it is quasicompact and Hausdor . X is said to be locally quasicompact if for all y 2 X and U1 2 with y 2 U1 , there exists a quasicompact subset Q and an open set U2 such that y 2 U2 and U2 Q U1 . X is coherent if it is sober, locally quasicompact and the nite intersection of 3
upper quasicompact sets is quasicompact. Let x; y be elements of a dcpo D. We say x approximates y or x is way below y, denoted by x y, if whenever y v F" A and A is a directed set, then there exists a 2 A with x v a. We have that x y implies that x v y . For C D we de ne "C = fx 2 D j 9c 2 C:c xg. We abbreviate "fcg by "c. In a similar way we de ne #C and #c. A subset B D is called F a basis if for all x 2 D the set Bx = B \ #x is directed and x = " Bx . D is a continuous dcpo if it has a basis and an !-continuous dcpo if it has a countable basis. In a continuous dcpo D for all C D the set "C is open. A continuous dcpo with the Scott topology is always sober, locally quasicompact but not necessarily coherent. Given a topological space (X; ), a valuation [4, 16, 14] is a map : ! [0; 1] which satis es:
(;) = 0 (strictness) Q Q ) (Q ) (Q ) (monotonicity) (Q [ Q ) + (Q \ Q ) = (Q ) + (Q ) (modularity). 1
2
1
1
2
2
1
2
1
2
A valuation is said to be continuous [16, 14] if for any D , which is S directed with respect to , we have ( D) = supQ2D (Q). For any a 2 X we de ne the point valuation based at a as the function a : ! [0; 1) such that (
a (Q) =
if a 2 Q otherwise.
1 0
A simple valuation is any nite linear combination ni=1 ri a of point valuations with ri 2 R+nf0g for i = 1; : : :; n. The set fa1 ; a2; : : :; ang is called the support of and is denoted by j j. Simple valuations are always continuous. A simple valuation can be extended to a measure on the whole power set of X and its extension to a measure on B(X ) is unique. For simplicity we will use the same name for a simple valuation and its extension. The probabilistic power domain PX of a topological space (X; ) is de ned as the set of continuous valuations on X bounded by 1 with the following order: v if and only if (Q) (Q) for all Q 2 . (PX; v) is a dcpo having the constant zero valuation as bottom element. For any directed F subset fi gi2I of PX , the lub = "i2I i is given by (Q) = supi2I i (Q) for all Q 2 . We denote by Ps X the subset of simple valuations of PX . The P
i
4
normalised probabilistic power domain of X is de ned as P1X = f 2PX j (X ) = 1g. If D is a dcpo with bottom element ? then P1D has ? as bottom element. Let J be a directed set and L be a set. A net is a function j 7! xj : J ! L. If L is a poset and j v k implies that xj v xk for all j; k 2 J then we say that the net is monotone. Nets will also be denoted as hxj ij 2J . Observe that the image of a monotone net is a directed subset of L. Conversely for any directed D L the inclusion map from D to L is a monotone net. We will use monotone nets and directed subsets interchangeably. Let hxj ij 2J be a net ranging over the real numbers. Recall that limj 2J xj = l 2 R if and only if for all " > 0 there exists k 2 J such that jxj ? lj < " if k v j . Let X be a set. A collection S of subsets of X is called a (Boolean) semialgebra of subsets of X if: (i) ;; X 2 S ; (ii) S is closed under nite intersections; (iii) if A 2 S then its complement Ac , can be expressed as a nite disjoint union of elements of S . If S is a semialgebra of subsets of X , then the set of nite disjoint unions of elements of S is the minimum algebra which contains S .
Lemma 2.1 (Splitting lemma) Let (D; ; v) be a monotone convergence P P space and = b2j1 j rb b , = c2j2 j sc c 2P D. Then v if and only if there exists a function f : j j j j ! [0; 1] such that for all b 2 j j and c 2 j j 1
1
2
1
2
1
2
1
2
c2j2 j f (b; c) = rb .
(1)
P
(2)
P
b2j1 j f (b; c) = sc .
(3) f (b; c) > 0 implies b v c.
Proof This result is known for dcpo's with the Scott topology [6, Proposition 3.1]. So we only need to note that j j [ j j with the inherited order 1
2
is a nite dcpo. The Scott topology in this space is the subspace topology induced by . Therefore, there exists f satisfying the above conditions. Conversely, if such f exists, then 1 v 2 with respect to D. Since D the conclusion follows. A general version of the above result exists for Polish spaces [25, Theorem 11] [15, Theorem 1-(ii)]. A subset T of a topological space (X; ) is called a crescent if there are open sets U and V such that V U and T = U nV . We denote the set of all crescents of (X; ) by Cres( ). The next Proposition is a reformulation of Lemma 1 of [23] for monotone convergence spaces. 5
Proposition 2.2 Let (D; ; v) be a monotone convergence space. (1) Cres( ). (2) T 2 Cres( ) if and only if there exist an open set A and a closed set B such that T = A \ B. (3) If T 2 Cres( ) then
{ If x; y 2 T and x v z v y then z 2 T . { If S T is a directed subset then F" S 2 T . { If S D is a directed subset and F" S 2 T then there exists d 2 S such that d 2 T . 0
0
(4) Cres( ) is a semialgebra. (5) Let : Cres( ) ! [0; 1] be a function. Then has a unique extension to a probability measure on the Borel -algebra of (D; ) if and only if is nitely additive, -subadditive and (D) = 1.
Proof (1) and (2) are trivial, (4) follows from (2). As a consequence we
have that the -algebra generated by Cres( ) is exactly the Borel -algebra of (D; ). (3) Assume that T = U nV with U; V 2 and V U . Recall D.
{ Suppose that x; y 2 T and x v z v y. We have z 2 U since x 2 U and U is upper. For the same reason if z 2 V then we would have y 2 V which is false. Therefore z 2= V and we conclude that z 2 T. F { Suppose that S T is a directed set. " S 2 U since U is upper. F If " S 2 V then there exists d 2 S such that d 2 V . This F F contradicts S T . Therefore " S 2= V and " S 2 T . { Since F" S 2 U and U is open then thereF exists d 2 S with d 2 U . If there exists d 2 S \ V then " S 2 V , since V is F" upper, contradicting S 2 T . Therefore S V c and d 2 T . 0
0
0
(5) Let : S ! [0; 1] be a function de ned on a semialgebra S and such that (;) = 0. Then has a unique extension to a measure on the algebra A generated by S if is nitely additive and -subadditive. By Caratheodory's extension theorem a - nite measure on an algebra 6
A has a unique extension to a measure on the -algebra generated by A [22, p. 295-298].
Lemma 2.3 (Smiley, Horn, Tarski) A normalised valuation on a topo-
logical space (X; ) extends uniquely to a nitely additive measure on the algebra of sets generated by the topology.
Proof A normalised valuation is a partial measure in the sense of Horn
and Tarski [11, Theorem 1.9]. Therefore it can be extended uniquely to a nitely additive measure on the Boolean algebra generated by the lattice of open sets (Theorem 1.22). See also [24]. Note that the algebra of sets generated by consists of nite disjoint unions of crescents. For any T 2 Cres( ) with T = U nV and V U we have (U nV ) = (U ) ? (V ).
Lemma 2.4 Let D be a continuous dcpo. Then D is !-continuous if and only if D is second countable.
Proof (cf. [19, Proposition 3.1]) For the `only if' part suppose that D is an
!-continuous dcpo with countable basis B. Then it is easy to see that the set f"b j b 2 B g is a countable basis for the topology. For the `if' part suppose that S is a countable basis for the topology. Consider the set P = f(u; v ) j u; v 2 S and 9w 2 D:u " w v g. Using a choice function we can construct a set B = fwp 2 D j p 2 P g such that for all w(u;v) 2 B we have u " w(u;v) v. Since P S S then B is a countable
set. We will show that B has the interpolation property [8, Chapter I,1.15], i.e. if x; z 2 D and x z then there exists w 2 B such that x w z. Suppose that x; z 2 D with x z . Since D is a continuous dcpo then there exists y 2 D such that x y z. Therefore y 2 "x and z 2 "y. It follows that there exist two basic open sets v; u 2 S such that y 2 v "x and z 2 u "y . Hence u "y " y v . As a consequence we have that (u; v ) 2 P . Therefore there exists w(u;v) 2 B such that z 2 u " w(u;v) v "x. We conclude that x w(u;v) z with w(u;v) 2 B . Since D is a continuous dcpo, checking that B is a basis is an easy consequence of the interpolation property. A valuation on a topological space (X; ) is called countably continuous S if for all ! -chains fQi gi2N in we have ( i2N Qi ) = supi2N (Qi ). 7
Lemma 2.5 Let D be an !-continuous dcpo. A valuation on D is continuous if and only if it is countably continuous.
Proof The `only if' part is immediate. For the `if' part let B = fbn j n 2 Ng
be a countable basis for the dcpo. Suppose that we have a directed union S Q = "k2K Qk where Qk 2 D for all k 2 K . We must show that (Q) = supk2K (Qk ). Since is a valuation, by monotonicity we have supk2K (Qk ) (Q). Let ci = fb1; b2; : : :; big for all i 2 N. Since B is a basis and Q is an open set, for all x 2 Q there exists m 2 N such that bm x and bm 2 Q. Hence S x 2 "bm "(Q \ cm). Therefore Q = i2N "(Q \ ci ) and f"(Q \ ci)gi2N is an ! -chain in D. Since we are assuming that is a countably continuous valuation we have (Q) = supi2N ("(Q \ ci )). Observe that for all i 2 N and S b 2 Q \ ci there exists a k 2 K such that b 2 Qk . Since the union "k2K Qk is directed we can nd ki 2 K such that Q \ ci Qk . Therefore for all i 2 N we have "(Q \ ci ) Qk . Hence ("(Q \ ci)) (Qk ) (Q) for all i 2 N. We conclude that supk2K (Qk ) = (Q). i
i
i
3 A generalised Splitting Lemma Let (D; ; v) be a monotone convergence space and fn gn2N be an ! -chain of normalised simple valuations on D. Consider a sequence of functions fi : ji j ji+1 j ! [0; 1] such that fi satis es the conditions of Lemma Q 2.1 for i and i+1 . Let S = i2N ji j endowed with the product topology of the discrete spaces ji j. For any A; B subsets of N with A B , let Q Q Q QB;A : i2B ji j ! i2A jij denote the natural projection of i2B ji j onto i2A ji j. For any E S and n 2 N de ne E n = N;f0;1;:::;ng (E ), i.e. the projection of the elements of E into their rst n +1 coordinates. For a given s = (x0; x1; : : :; xn) 2 S n we de ne
pn (fsg) = 0 (fx0 g)
fi (xi ; xi+1 ) : i (fxi g) 0i 0 only if x v y , it follows that P (C ) = 1.
Proposition 3.2 Let fngn2N be an !-chain in Ps D. Let (An) be a sequence of pairwise disjoint sets such that An jn j. Suppose that for some xed > 0, n (An ) , for all n 2 N. Then there exists (xk ) 2 C such that xn 2 An for in nitely many n 2 N. 1
9
Proof Let C (An) = C \ hn; Ani, i.e. the set of chains (xk ) 2 S such that xn 2 An. Since C is closed, it is Borel measurable, therefore P (C (An )) = P (C \ hn; Ani) = P (hn; An i) since P (C ) = 1 = n (An ) : Now observe that the subset of chains (xk ) 2 C such that xn 2 An for T S in nitely many n 2 N, is exactly m2N nm C (An ) = lim supn C (An ). Therefore P (lim supn C (An )) > 0. Since P is a measure we conclude that lim supn C (An ) 6= ;. This last result also follows from extended versions of the Borel-Cantelli Lemma for nonindependent events (see [17, Lemma 3.5] or [18]).
4 The extension results We rst prove the extension theorem in the case of normalised simple valuations.
Theorem 4.1 Let (D; ; v) be a monotone convergence space and hj ij2J be a directed subset of Ps D with = supj 2J j . Then has a unique 1
extension to a measure on the Borel -algebra of (D; ).
Proof Since is the supremum of a directed family of continuous valuations,
it is easy to check that is a continuous valuation and (D) = 1. As a consequence of Lemma 2.3, extends uniquely to a nitely additive set function de ned on Cres( ). In order to apply part (5) of Proposition 2.2 we only need to check that is -subadditive on Cres( ) . Suppose that T 2 Cres( ) is the union of a countable disjoint sequence P (Ti ) in Cres( ). We show that (T ) 1 i=1 (Ti). In order to derive a Pm P1 contradiction assume limm i=1 (Ti) = i=1 (Ti) < (T ). This happens if and only if there exists some > 0 such that for all m 2 N we have S Pm i=1 (Ti) (T ) ? 2 . Therefore 2 limj j ( i>m Ti ). This means that if we x m 2 N there exists nm 2 J such that nm v j implies that S j ( i>m Ti) . Let N0 2 J be such that N0 (T ) . Observe that for all j 2 J , the set jj j \ T is nite. SIn particular since jN0 j is nite, there existsSp1 2 N such that jN0 j \ T mp1 Tm . Since we are assuming limj 2J j ( m>p1 Tm ) 10
hj ij2J 2 , there exists N1 2 J such that N1 ( m>p1 Tm ) . Because is directed we can also assume N0 v N1 . In general, since N \ T S mp +1 Tm is nite we can repeat the argument and nd Nj +1 2 J and pj+2 2 N such that S
j
j
N v N +1 N +1 \ T Smp +2 Tm Aj = N +1 \ Sp +1 0 , let 0 = =(D) and 0j = j =(D). Then 0 (D) = 1 and 0 = supj 2J 0j where h0j ij 2J is a directed set of simple valuations. It is clear that will have a unique extension to a measure if and only if 0 does. If D has a bottom element ? then for all j 2 J , let
j = 0j + (1 ? 0j (D)) ?: Then j is a simple valuation, j (D) = j (" ?) = 1 and j (Q) = 0j (Q) for all Q 2 nfDg. Therefore hj ij 2J is a directed subset of P1s D with 0 = supj2J j and the conclusion follows from Theorem 4.1. 11
If D does not have a bottom element we can consider the lifting D? = D [ f?g where ? is the least element of D? . Then we have ? = [fD? g. Since 0 (D) = 1, it is clear that we can extend 0 to D? by letting 0 (D?) = 1. The de nition given above for the j will do as an extension for the 0j . Now we can apply Theorem 4.1 and nd a unique extension of 0 to a measure 0 on B(D? ). The restriction of this measure to B(D) will give us the desired measure since 0 (?) = 0(D? nD) = 0 (D? ) ? 0 (D) = 0.
Corollary 4.3 If D is a continuous dcpo then every bounded valuation has
a unique extension to a measure on the Borel -algebra of (D; D).
Proof If D is a continuous dcpo then the set of continuous valuations
on D is also a continuous dcpo with a basis of simple valuations [14, 13]. Hence every bounded valuation is the supremum of a directed set of simple valuations and the conclusion follows from the last corollary. This shows that the extension results considered in this paper are not directly implied by the extension theorem of Norberg and Vervaat. A continuous dcpo need not be coherent and the latter theorem is given for spaces where the intersection of upper quasicompact sets is quasicompact. On the other hand, we do not know if a continuous valuation on a coherent space is always the lub of a directed set of simple valuations. Hence, our results might not apply in that setting. Similar remarks apply to Lawson's result for locally quasicompact second countable sober spaces.
5 The - nite case Next we present the extension result for - nite valuations.
Theorem 5.1 (D; ; v) be a monotone convergence space. Let hjSij2J be a directed set of simple valuations on D with = supj 2J j . If D = i2N Qi with Qi 2 and (Qi ) < 1 then has a unique extension to a measure on the Borel -algebra of (D; ).
Remark 5.2 This theorem is only more general in case D is not quasicompact. Otherwise, under the assumptions of the theorem, fQi gi2N would have a nite subcover, implying (D) < 1 and Theorem 4.1 would apply. In particular, this is true if D =" F for some nite set F D. Proof of Theorem 5.1 Without loss of generality we can assume Qi Qi for all i 2 N. As in the proof of Theorem 4.1 we will rst extend +1
12
to the semialgebra Cres( ) and show that this extension is nitely additive and -subadditive. The idea is to extend to a nite measure on Qi and then take the supremum of such measures. Let be a continuous valuation on D and Q 2 . The restriction of to Q de ned as jQ (A) = (A \ Q) for all A 2 is also a continuous valuation P P on D. If = b2j j rbb is a simple valuation then jQ = b2j j\Q rb b is also a simple valuation. It is straightforward to check that hj jQij 2J is a directed set of simple valuations on D and jQ = supj 2J j jQ . Notice that for all i 2 N, jQ (D) = (Qi ) < 1; hence by Corollary 4.2 it has a unique extension i to a Borel measure on (D; ). Let T = U nV 2 Cres( ) with U; V 2 and V U . We de ne the extension of to Cres( ) by (T ) = supi2N i (T ). The sequence (i(T )) is increasing since for all i 2 N we have i
i (T ) = i(U ) ? i(V ) = jQ (U ) ? jQ (V ) = limj 2J j jQ (U ) ? limj 2J j jQ (V ) = limj 2J j (T \ Qi ): i
i
i
i
But Qi Qi+1 implies that j (T \ Qi ) j (T \ Qi+1 ) for all j 2 J and hence i (T ) = lim j 2J j (T \ Qi ) lim j 2J j (T \ Qi+1 ) = i+1 (T ). So we F can write (T ) = "i2N i (T ). Next, is an extension of since for all A 2 we have
(A) =
" (A) i2N i F" i2N jQi (A) since A 2 F" iS 2N (A \ Qi ) ( i2N A \ Qi ) since is continuous S (A \ i2N Qi ) F
= = = = = (A):
We can easily check that is nitely additive and -subadditive, therefore has a unique extension to a Borel measure. We can now give a direct proof of a result by Norberg.
Corollary 5.3 [19, Theorem 3.9] Let D be a continuous dcpo with a second countable Scott topology. A function : D ! [0; 1] that is nite on the set fU 2 D j U Dg has a unique extension to a Borel measure if and only if it is a countably continuous valuation.
13
Proof The `only if' part is immediate since the restriction of a Borel measure
to the open sets gives a countably continuous valuation. For the `if' part notice that since D is a continuous dcpo with second countable Scott topology then, by Lemma 2.4, D is an ! -continuous dcpo. In an ! -continuous dcpo countably continuous valuations are continuous by Lemma 2.5. Hence is a continuous valuation. Moreover, on a continuous dcpo the set of continuous valuations is a continuous dcpo with a basis of simple valuations [14, 13]. Therefore is the supremum of a directed set of simple valuations. Let B = fbn j n 2 Ng be a countable basis for D. We have "bn D for all n 2 N. Hence ("bn ) < 1 for all n 2 N and S D = n2N "bn. The conclusion follows from Theorem 5.1. Suppose, more generally, that is a continuous valuation on a monotone convergence space (D; ; v) and (D) = 1. If D has a bottom element ? then the set f?g is closed. Therefore D0 = Dnf?g 2 . Hence (D0 ) < 1 or (D0 ) = 1. In the rst case it is straightforward to check that jD = supj 2J j jD ; then by Corollary 4.2 jD can be extended uniquely to a Borel measure on D0. In order to extend this measure to the whole space D we only need to assign some weight to f?g. Since a measure is nitely additive and we are assuming (D0 ) < 1 the only possible choice is to have (f?g) = 1. So if (D0) < 1 the extension is unique. If (D0 ) = 1 and jD has a unique extension to a Borel measure on D0 , (f?g) can be assigned any nonnegative extended real value and this function will still be a measure; therefore the extension to D is not unique in this case. 0
0
0
0
6 A counterexample for a dcpo The question remains if the extension result holds in more general settings. We will give an example of a bounded continuous valuation on a dcpo which cannot be extended to a Borel measure. The following dcpo is de ned in [12, Chapter II, 1.9], as an example of a dcpo with a non-sober Scott topology. Consider X = N (N [f1g) with (j; k) v (m; n) if and only if either j = m and k n or n = 1 and k m. It is easy to check that (X; v) is a dcpo and that every nonempty open set contains all but a nite number of points (m; 1). De ne the function : X ! [0; 1] by (
(Q) =
if Q 6= ; otherwise.
1 0 14
Then is a modular function since the intersection of any pair of nonempty open sets in X is again nonempty. Strictness, monotonicity and continuity are easily veri ed, therefore is a bounded continuous valuation. But S cannot be extended to a Borel measure. In fact Qn = ( nj=0 # (j; 1))c is a T decreasing sequence of open sets with n2N Qn = ; but limn (Qn ) = 1. By Theorem 4.1 also gives an example of a continuous valuation on a dcpo that is not the supremum of a directed family of simple valuations.
Acknowledgement This work started after O. Kirch and R. Tix pointed out the gaps in [23] and [13]. We would like to thank Philipp Sunderhauf for useful comments on a preliminary version of this paper. We thank the anonymous referee who suggested the generalisation of the extension result from dcpo's to monotone convergence spaces. Jimmie Lawson provided the short proof of Lemma 2.4. Klaus Keimel made us aware of [20]. The third author acknowledges the hospitality of the Department of Computing at Imperial College during his academic visit to London in 1998. The rst author was supported by Universidad Nacional Autonoma de Mexico (UNAM). The second author was supported by EPSRC, UK.
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[6] A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163{193, 1995. [7] A. Edalat. Domains for computation in mathematics, physics and exact real arithmetic. The Bulletin of Symbolic Logic, 3(4):401{452, 1997. [8] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A compendium of continuous lattices. Springer-Verlag, Berlin, 1980. [9] Paul R. Halmos. Measure Theory. D. Van Nostrand Company, Princeton, New Jersey, 1950. [10] R. Heckmann. Spaces of valuations. In S. Andima, R.C. Flagg, G. Itzkowitz, Y. Kong, R. Kopperman, and P. Misra, editors, Papers on General Topology and Applications: eleventh summer conference at the university of southern maine, volume 806, New York, New York, 1996. Annals New York Academy of Sciences. [11] A. Horn and A. Tarski. Measures in Boolean algebras. Trans. Amer. Math. Soc., 64:467{497, November 1948. [12] P. Johnstone. Stone spaces. Cambridge University Press, UK, 1982. [13] C. Jones. Probabilistic Non-determinism. PhD thesis, University of Edinburgh, 1989. [14] C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Logic in Computer Science (LICS), pages 186{195. IEEE Computer Society Press, Silver Spring, MD, 1989. [15] T. Kamae, U. Krengel, and G. L. O'Brien. Stochastic inequalities on partially ordered spaces. The Annals of Probability, 5(6):899{912, 1977. [16] J.D. Lawson. Valuations in continuous lattices. In Rudolf-Eberhard Ho mann, editor, Continuous Lattices and Related Topics, volume 27 of Mathematik Arbeitspapiere. Universitat Bremen, 1982. [17] M. Loeve. On almost sure convergence. In Procceedings of the Second Berkeley Symposium on Mathemathical Statistics and Probability, pages 279{303. University of California Press, 1951. [18] S. W. Nash. An extension of the Borel-Cantelli Lemma. Annals of mathematical statistics, 25:165{168, 1954. 16
[19] T. Norberg. Existence theorems for measures on continuous posets, with applications to random set theory. Math. Scand., 64:15{51, 1989. [20] Tommy Norberg and Wim Vervaat. Capacities on non-Hausdor spaces. In Wim Vervaat and Henk Holwerda, editors, Probability and lattices, number 110 in CWI Tracts. SMC, Centrum voor Wiskunde en Informatica, Amsterdam, Holland, 1997. [21] B.J. Pettis. On the extension of measures. Annals of Mathematics, 54(1):186{197, 1951. [22] H.L. Royden. Real Analysis. Macmillan Publishing Company, New York, NY, third edition, 1988. [23] N. Saheb-Djahromi. Cpo's of measures for nondeterminism. Theoretical Computer Science, 12:19{37, 1980. [24] M. F. Smiley. An extension of metric distributive lattices with an application in general analysis. Trans. Amer. Math. Soc., 56:435{447, 1944. [25] Volker Strassen. The existence of probability measures with given marginals. The Annals of Mathematical Statistics, 36:423{439, 1965. [26] R. Tix. Stetige Bewertungen auf topologischen Raumen. Master's thesis, Technische Hochschule Darmstadt, 1995.
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M.A. Department of Computing Imperial College of Science, Technology and Medicine 180 Queen's Gate. London SW7 2BZ. UK. e-mail:
[email protected] A.E. Department of Computing Imperial College of Science, Technology and Medicine 180 Queen's Gate. London SW7 2BZ. UK. e-mail:
[email protected] N.S. LaBRI Universite Bordeaux-I 351 cours de la Liberation 33405 Talence, France. e-mail:
[email protected]
18
LIST OF SYMBOLS
A B
N R
S "
2
v ! w
? " # v F
"
Script capital A Script capital B Open face capital N Open face capital R Script capital S lowercase epsilon belongs to lowercase nu lowercase v lowercase omega lowercase w tau restriction symbol way below relationship end of proof symbol bottom element symbol double up arrow double down arrow subset of dominated by Directed supremum (note that arrow is superscripted)
19
