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Riemann and Edalat integration on domains

2003, Theoretical Computer Science

The main result of this paper is that the domain-theoretic approach to the generalized Riemann integral ÿrst introduced by Edalat extends to a large class of spaces that can be realized as the set of maximal points of domains.

Theoretical Computer Science 305 (2003) 259 – 275 www.elsevier.com/locate/tcs Riemann and Edalat integration on domains Jimmie D. Lawsona;∗ , Bin Lub a Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA b University of Arizona, Tucson, AZ 85721, USA Abstract The main result of this paper is that the domain-theoretic approach to the generalized Riemann integral rst introduced by Edalat extends to a large class of spaces that can be realized as the set of maximal points of domains. The approach is based on the theory of a Riemann–Stieltjes type integral on a topological space with respect to a nitely additive measure. We develop the theory of this integral for a bounded function f de ned on the maximal points of a continuous domain and show that it gives an alternate approach to the Edalat integral. c 2002 Elsevier B.V. All rights reserved.  Keywords: Riemann integral; Continuous domain; Finitely additive measure; Algebra of sets; Valuation 1. Introduction The domain-theoretic approach of generalizing Riemann integration was rst introduced by Edalat [5,6]. Edalat introduced the continuous domain of non-empty compact subsets, called the upper space, of a compact metric space and established that the normalized Borel measures on a compact metric space can be identi ed with the maximal points of the probabilistic power domain of the upper space with the Scott topology [5]. Moreover, from the theory of the probabilistic power domain, each measure can be approximated by a chain of simple valuations, which play the role of partitions in the classical theory. With the help of this chain of simple valuations, the generalized Darboux sums and Riemann integral can be introduced. Edalat showed that this integral, which we call the Edalat integral or E-integral, preserves many standard properties of the classical Riemann integral. It can furthermore be successfully applied to a variety of computations such as those arising in fractal geometry and ∗ Corresponding author. Tel.: +1-225-578-1672; fax: +1-225-578-4276. E-mail addresses: [email protected] (J.D. Lawson), [email protected] (B. Lu). c 2002 Elsevier B.V. All rights reserved. 0304-3975/03/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 6 9 6 - 5 260 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 stochastic processes. Indeed, the desire for approaches to integration that suggest e ective computational algorithms was a major motivation for the introduction of this integral. In a recent paper [11] Howroyd has shown that the Edalat integral can be directly and readily extended to a large class of spaces that includes those that may be realized as the maximal point spaces of continuous domains (Edalat and Negri had extended Edalat’s approach to locally compact spaces [10], but not beyond). Independently, the present authors had carried out an overlapping program. However, our approach and techniques are quite di erent from those of Howroyd. We apply the theory of nitely additive measures to monotonic extensions of bounded functions on the set of maximal points of a domain. We show that the integral that we obtain via this approach agrees with that of Howroyd, and hence also that of Edalat. 2. The Riemann–Stieltjes integral for algebras of sets In this section we introduce some basic concepts and results concerning integrals for algebras of sets. The theory so closely resembles that of classical Riemann integration on intervals that we only sketch the development. Since Stieltjes suggested the extension of the Riemann integral to more general measures or masses (on the real line), it seems appropriate to call these Riemann–Stieltjes integrals for algebras of sets. Details may be found in Chap. 4.5 of [3], where such integrals are called S-integrals for short, a terminology that we also adopt. Throughout this section let X denote a non-empty set, A an algebra of subsets of X (closed under nite unions, nite intersections, and complements), and  a positive bounded and nitely additive measure on A. We always assume implicitly throughout the paper that all measures are non-negative and bounded. An A-partition of X is a partition by subsets all of which belong to A. De nition 2.1. Let f : X → R be a bounded function. The Darboux upper sum of f with respect to  and P, a nite A-partition of X , is de ned by S u (f; ; P) =  sup f(P)(P): P∈P Similarly the Darboux lower sum of f with respect to  and P is de ned as S l (f; ; P) =  inf f(P)(P): P∈P Since f and  are bounded, the upper and lower sums, S u (f; ; P) and S l (f; ; P), are well-de ned. The following proposition follows in a straightforward manner from the previous de nition. J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 261 Proposition 2.2. Let P1 and P2 be two nite A-partitions. If P1 re nes P2 in the sense that every set in P2 is a union of sets in P1 , then we have the following inequalities: S l (f; ; P2 ) 6 S l (f; ; P1 ) 6 S u (f; ; P1 ) 6 S u (f; ; P2 ): From these lower and upper sums we de ne the lower and upper integrals accordingly. De nition 2.3. The upper integral of f with respect to  is de ned as  − f d = inf S u (f; ; P); P where P is a nite A-partition of X . Similarly the lower integral is de ned as  f d = sup S l (f; ; P): P − De nition 2.4. A bounded function f : X → R is said to be S-integrable if   − f d: f d = − If f is S-integrable, the integral of f is denoted by S value of the lower or upper integral, i.e.,  −   S f d: f d = f d = X  f d, and is de ned to be the − Remark 2.5. Suppose that S is a semialgebra (a non-empty collection of subsets closed under nite intersections and having that property that for S ∈ S, the complement S c is a nite disjoint union of members of S). Then the collection A consisting of the empty set and all nite disjoint unions of members of S is an algebra, the smallest algebra of sets containing S, called the algebra generated by S. Given, a nitely additive bounded measure  on A, one can consider only S-partitions and de ne S-integrals with respect to S in a manner entirely analogous to the way they are de ned for an algebra of sets. Furthermore, since any nite A-partition may be re ned to a nite S-partition, it readily follows from Proposition 2.2 that a bounded real-valued function f is S-integrable with respect to S if and only if it is S-integrable with respect to A. Thus, the whole theory applies equally well to semialgebras as to algebras, with the same theory arising for a semialgebra and the algebra it generates, although we present our results in the algebra context. Remark 2.6. One can show without great diculty that the classical theory of Riemann integration on intervals of the real line agrees with that given here for the semialgebra of half-open, half-closed intervals (a; b], and hence the theory developed here may be 262 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 viewed as a generalization of classical Riemann integration. Similar remarks apply to classical Riemann–Stieltjes integration. The following are further results collected from [3]. The rst two are straightforward generalizations from classical Riemann integration theory. Theorem 2.7. Let f : X → R be a bounded function. Then the following are equivalent: (1) f is S-integrable. (2) There exists a real number I with the following property: for every ¿0 there exists a nite A-partition P = {A1 ; : : : ; An } of X such that for every xi ∈ Ai , i = 1; : : : ; n,   n    f(xi )(Ai ) − I  ¡    1 holds. (3) For every ¿0 there exists a nite A-partition P = {A1 ; : : : ; An } such that for every choice xi ; yi ∈ Ai , i = 1; : : : ; n,   n    (f(xi ) − f(yi ))(Ai ) ¡    1 holds. (4) For every ¿0, there exists a nite A-partition P = {A1 ; : : : ; An } such that n  sup |f(xi ) − f(yi )|(Ai ) ¡ : 1 xi ;yi ∈Ai  In case (2) the real number I is unique and is equal to S X f d. In cases (2) through (4) the inequality continues to hold for any partition that re nes P. Theorem 2.8. If f and g are S-integrable, a ∈ R, then af and f + g are S-integrable and        S g d: f d + S (f + g) d = S f d ; S af d = a S X X X X X Theorem 2.9. A bounded function f : X → R is S-integrable if and only if for each  ¿0, there exist simple A-measurable functions g; h such that h6f6g and S X (g − h) d¡. Proof. Let ¿0 and pick (by the de nition of S-integrability) A-partitions P1 and P2 such that    inf f(P)(P) ¡ f d − S 2 P∈P1 X J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 263 and  sup f(P)(P) − S P∈P2  f d ¡ X  : 2 By Theorem 2.7 the same inequalities hold for an A-re nement P of P1 and P2 . For each A ∈ P, de ne g on A to be sup f(A) and h on A to be inf f(A). Clearly g and h are A-measurable simple functions and h6f6g. For the partition P and the functionh, condition (2) of Theorem 2.7 is satis ed for every ¿0 if I is chosen to be and hence the latter is the S-integral; a similar result P∈P inf f(P)(P),  holds for g. That S X (g − h) d¡ now follows from the choice of P. Since we are only interested in the preceding implication, we leave the converse as a straightforward exercise for the reader. Theorem 2.10. Suppose that f : X → R is a bounded and upper measurable, i.e., f−1 ((a; ∞)) ∈ A for all a ∈ R, or lower measurable, i.e., f−1 ((−∞; a)) ∈ A for all a ∈ R with respect to A. Then f is S-integrable. Proof. Partition an interval containing the range into small (¡=(X )) appropriately half-open, half-closed intervals and apply condition (4) of Theorem 2.7. Corollary 2.11. Let f : X → R be a bounded function such that for every ¿0, there exists A ∈ A such that (A)¡ and f is upper (resp. lower) measurable on the complement Ac of A. Then f is S-integrable. Proof. Suppose that |f(x)|6B for all x ∈ X . Pick A such that (A)¡=4B and f is upper measurable on Ac . By the preceding theorem n and (4) of Theorem 2.7 there exists an A-partition {P1 ; : : : ; Pn } of Ac such that i=1 |f(xi ) − f(yi )|(Pi )¡=2 for every choice xi ; yi ∈ Pi for i = 1; : : : n. Then adding A to the partition gives a partition of X that satis es condition (4) of Theorem 2.7. A set G ⊆ X is called a null set if for every ¿0, there exists A ∈ A such that G ⊆ A and (A)¡. Two functions f and g are equal almost everywhere (a.e.) if they di er on a null set. Corollary 2.12. If bounded real-valued functions f and g are equal a.e. and f is S-integrable, then g is S-integrable and the two integrals agree. Proof. Since g − f is equal to 0 a.e., it follows easily from the preceding corollary that g−f is S-integrable, and then it follows directly from the de nition of the integral that it must be 0. Then     f d: (g − f) d = S f d + S g d = S S X X X X 264 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 De nition 2.13. For X a topological space, let A be the smallest algebra containing the semialgebra of crescent sets, i.e., sets of the form U \V where U; V are open subsets of X . By replacing V with U ∩ V , we may always assume without loss of generality that V ⊆ U . The algebra A is called the crescent algebra. The Scott topology on R consists of R, the empty set, and all open right rays (a; ∞). A function is continuous a.e. if the set of points on which it is discontinuous is a null set. Theorem 2.14. If X is a topological space and a bounded function f : X → (R; Scott) is continuous a.e., then f is integrable with respect to the crescent algebra of open sets, and hence also with respect to the semialgebra of crescent sets. Proof. For each ¿0, we can nd A ∈ A such that (A)¡ and f is continuous for the Scott topology, and hence upper measurable, on Ac . Thus by Corollary 2.11, f is integrable. Finally we consider the case that A is a -algebra on a set X and  is a countably additive bounded measure (always assumed non-negative) de ned on A. Suppose that f : X → R is bounded and S-integrable with respect to the -algebra. Then using Theorem 2.9 (and the construction in its proof) we obtain inductively for each n a nite A-partition Pn re ning Pn−1 and simple functions gn and hn with level sets the members of Pn such that hn−1 6hn 6f6gn 6gn−1 and S X (gn −hn ) d¡1=n. Set g = inf n gn and h = supn hn . Then g and h are Lebesgue integrable Dominated    from the Lebesgue  Convergence Theorem, h6f6g, and h d = sup h d and g d = inf n X gn n X n X  X   d. It follows that S X f d = X h d = X g d. Thus g − h¿0 and has integral 0, and thus must be non-zero on a null set. Since it dominates f − h, it follows that f is equal to h (and hence g), a.e. We summarize: Theorem 2.15. Let A be a -algebra on a set X and  be a countably additive bounded measure (always assumed non-negative) de ned on A. A S-integrable function f is almost measurable, that is, equal to a measurable function a.e., indeed to one that is integrable, and its S-integral is equal to the Lebesgue integral of any such function. In particular, if f is measurable (which will be the case if the measure is complete) and S-integrable, then it is Lebesgue integrable and the two integrals agree. 3. Basic domain theory In this section we quickly recall basic notions concerning continuous domains (see [1]). A subset D of a partially ordered set (X; ⊑) is directed if given x; y ∈ D, there exists z ∈ D such that x; y ⊑ z. A directed complete partially ordered set or dcpo is a partially ordered set (X; ⊑) such that every directed subset of X has a least upper bound in X . J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 265 Let x; y ∈ X where X is a dcpo. Then we say x approximates y, denoted by x ≪ y, if for every directed set D with y ⊑ sup D we have x ⊑ d for some d ∈ D. For y ∈ X we de ne ⇓ y = {x ∈ X : x ≪ y}: Then we say a dcpo X is continuous if • y = sup ⇓ y for all y ∈ X and • each ⇓ y is a directed set. A base for a continuous dcpo X is a set B ⊆ X such that for all x ∈ X , x = sup{⇓ x ∩ B}; and the supremum is taken over a directed set. A domain is a continuous dcpo and an !-continuous domain is a domain with a countable base. For a dcpo X , we can de ne the Scott topology as follows: a subset O ⊆ X is Scott open if • O is an upper set, i.e., if x ⊑ y and x ∈ O, then y ∈ O. • O is inaccessible by least upper bounds of directed sets, i.e., if sup D ∈ O for a directed set D, then d ∈ O for some d ∈ D. A function between dcpos X and Y is Scott continuous if it is monotone and preserves directed suprema. Equivalently a Scott continuous function is continuous with respect to the Scott topologies on X and Y . Every continuous domain has associated with it a probabilistic power domain that allows one to make interesting connections between measure and integration theory and applications thereof (see [5,6]). De nition 3.1. A valuation on a topological space X is a function  : X → [0; ∞), where X is the set of all open subsets of X , which for all U; V satis es: (i) (U ) + (V ) = (U ∩ V ) + (U ∪ V ), (ii) (∅) = 0, (iii) U ⊆ V implies (U )6(V ). Valuations have a natural pointwise order given by 6 if (U )6(U ) for all open sets U . A continuous valuation is a valuation such that whenever D is a directed family with respect to inclusion in X then     O = sup (O): O∈D O∈D A point valuation a based at a is a continuous valuation de ned as follows:  1 if a ∈ O; a (O) = 0 otherwise: Any nite linear combination of point valuations n i=1 ri ai is called a simple valuation. 266 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 De nition 3.2. The probabilistic power domain PX of a topological space X consists of the set of continuous valuations  on X with (X )61 and with the pointwise order and the subset of normalized valuations, those with (X ) = 1, is called the normalized probabilistic power domain and denoted P 1 X . The following basic result due to Jones appears in [12]: Theorem 3.3. If X is an (!)-continuous dcpo, then the (normalized) probabilistic power domain is also (!)-continuous and has a basis consisting of simple valuations. It is clear that a measure on X becomes a valuation when restricted to (X ). Conversely by the Smiley–Horn–Tarski Theorem (see, for example, [14]), any valuation on the lattice of open sets extends uniquely to a nitely additive measure on the smallest algebra containing the open sets. There have been several attempts to extend a given continuous valuation to a countably additive measure (see, for example, [15]). The following general extension result has been proved recently (see details in [2]). Theorem 3.4 (Alvarez–Manilla et al. [2]). If D is a continuous domain then every bounded continuous valuation has a unique extension to a measure on the Borel -algebra of the Scott topology. 4. The S-integral for order-preserving functions In this section we assume that D is a continuous domain equipped with the Scott topology. Let A denote the crescent algebra generated by the Scott open sets ( nite disjoint unions of crescents). Then, as previously noted, by the Smiley–Horn–Tarski Theorem any valuation  on the Scott open sets extends uniquely to a nitely additive measure on A, which we continue to denote as . The results we derive in this section are valid for both the probabilistic power domain and the normalized probabilistic power domain. For a ∈ D, we de ne the point measure a on A by a (A) = 1 is a ∈ A and a (A) = 0 otherwise. Note that this extends the valuation that arises by applying the point measure to the lattice of Scott open sets, also denoted na . Corresponding to a simple valuation  = i=1 ri ai , where ri ¿0 and ai ∈ D, we also have the corresponding on A. For f : D → R we also de ne the corre simple measure n sponding integral by X f d = i=1 ri f(ai ) (note that this need not be the S-integral). De nition 4.1. A function f : D → R is order-preserving if and only if x ⊑ y implies f(x)6f(y) for all x; y ∈ D. Lemma 4.2. Suppose that f : D → R is a bounded order-preserving function. Let 1 ; 2 be two simple valuations on D with 1 ⊑ 2 . Then we have   l S (f; 1 ; P1 ) 6 f d2 6 S u (f; 2 ; P2 ); f d1 6 D D J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 267 where P1 ; P2 are two A-partitions of D and S l (f; 1 ; P1 ); S u (f; 2 ; P2 ) are the Darboux lower and upper sums, respectively. Proof. From Theorem 2.7, it suces to prove that result for P := P1 ∧P2 , the common re nement of the two. Let P = {A1 ; : : : ; An }, where each Ai belongs to A. Then S l (f; 1 ; P) = n  inf f(Ai )1 (Ai ) =  b∈|1 | = inf f(Ai ) i=1 i=1 = n   rb n   rb b (Ai ) b∈|1 |  inf f(Ai )b (Ai ) = i=1 rb inf f(Ab )b (Ab ) b∈|1 | rb inf f(Ab ) 6  rb f(b) = b∈|1 | b∈|1 |  f d1 ; D where Ab is the partition set such that b ∈ Ab . Similarly,   f d2 ; sc sup f(Ac ) ¿ S u (f; 2 ; P) = D c∈|2 | where Ac is the partition set such that c ∈ Ac . Since f is order-preserving function, we have f(b)6f(c) if b ⊑ c. On the other  hand, by the Splitting Lemma (see [12] or  [6]), there exists tb; c ¿0, such that rb = c∈|2 | tb; c and b∈|1 | tb; c 6sc (in the case of the normalized probabilistic power domain the last inequality is an equality), and tb; c ¿0 if and only if b ⊑ c. Therefore,  f d1 = D  rb f(b) =   tb;c f(b)   tb;c f(b)  f(c)  f(c)sc = b∈|1 | c∈|2 | b∈|1 | = c∈|2 | b∈|1 | 6 c∈|2 | 6 c∈|2 |  tb;c b∈|1 |  f d2 : D The lemma now follows by combining the previous results. As we know from Theorem 3.3, the probabilistic power domain of a continuous domain is also a continuous domain with a basis of simple valuations. Thus a continuous ↑ i , where each i is a simple valuation. From valuation  can be written as  = this we can compute the S-integral in terms of the integrals with respect to simple valuations. Throughout the remainder of this section the S-integral is computed with respect to the algebra A generated by the Scott open sets. 268 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 ↑ Theorem 4.3. Let  be a continuous valuation and  =  , where { } is a directed family of simple valuations. If f : D → R is bounded, order-preserving, and S-integrable with respect to , then   f d ; f d = sup S D D where the right-hand supremum is over the directed set. Proof. It follows directly from Lemma 4.2 that the right-hand integrals are directed. Let |f(x)|6M for all x. Pick an A-partition P = {A1 ; : : : ; An } such that   n n   inf f(Aj )(Aj ) ¡ : f d − f d ¡  and S sup f(Aj )(Aj ) − S j=1 j=1 D D Since each member of the algebra A is a nite union of crescents U \V , V ⊆ U , U and V Scott open, we may assume without loss of generality that Aj = Uj \Vj for each j (see Remark 2.5). Since  is the directed supremum of the { }, (Uj ) resp. (Vj ) is the directed supremum of  (Uj ) resp.  (Vj ) for all j = 1; : : : ; n. Thus, there exists an index such that |(Uj ) −  (Uj )|¡ and |(Vj ) −  (Vj )|¡, where  := =2Mn, for j = 1; : : : ; n. Then for all j, |(Aj ) −  (Aj )| = |(Uj ) − (Vj ) −  (Uj ) +  (Vj )| 6 |(Uj ) −  (Uj )| + |(Vj ) −  (Vj )| ¡ =Mn: Thus, S  f d − D  f d 6 S D  f d − D n  inf f(Aj ) (Aj ) (Lemma 4:2) j=1      n    inf f(Aj )(Aj ) f d − 6 S   D j=1     n    n inf f(Aj ) (Aj ) +  inf f(Aj )(Aj ) −  j=1 j=1 ¡+ n  | inf f(Aj )| |(Aj ) −  (Aj )| j=1 ¡+ n   M = 2: Mn j=1 Similarly,    n  f d sup f(Aj )(Aj ) − S f d 6 f d − S D D j=1 D J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 269     n    n sup f(Aj )(Aj ) 6  sup f(Aj ) (Aj ) −  j=1 j=1 ¡        n f d +  sup f(Aj )(Aj ) − S  j=1 D n  | sup f(Aj )| | (Aj ) − (Aj )| +  j=1 ¡ n   M +  = 2: Mn j=1     Thus, |S D f d  − D f d |¡2. This proves that lim D f d = S D f d, and since the set { D f d } is directed, it must converge to its supremum. Corollary 4.4. Let 1 and 2 be two continuous valuations on a continuous domain D with 1 ⊑ 2 . If a bounded f : D → R is S-integrable with respect to 1 and 2 and is order-preserving, then   S f d2 : f d1 6 S D D ↑ ↑ Proof. Let 1 = 1i , 2 = 2i , where {1i } and {1i } are families of simple valuations, which approximate 1 and 2 , respectively. Since 1 ⊑ 2 , for each i, ↑ 1i ≪ 1 ⊑ 2 = 2i . This implies that 1i ≪ 2 . Then from the de nition of approximation, for each i there exists k(i) such that 1i ⊑ 2k(i) . Hence by Lemma 4.2, we have   f d2k(i) ; f d1i 6 D D for each i. Hence by Theorem 4.3, we are done. In classical Riemann integration theory, the Riemann integral of a real-valued function can be approximated by the corresponding Darboux upper and lower sums. In the domain setting we have a similar result. Theorem 4.5. Let f : D → R be bounded, order-preserving, and S-integrable with re↑ spect to . Suppose that  = i , where {i : i ∈ J }, J an index set, and i is a simple valuation for each i. Then for all ¿0 there exist an A-partition P of D and an index k such that for all j¿k   f d + : f d −  ¡ S l (f; j ; P) 6 S u (f; j ; P) ¡ S S D D 270 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 Proof. Since f is S-integrable, then for each ¿0, there exists P, an A-partition of D, such that   f d + : f d −  ¡ S l (f; :P) 6 S u (f; ; P) ¡ D D Using the methods of the proof of Theorem 4.3, we can make S l (f; i ; P) arbitrarily close to S l (f; ; P) and S u (f; i ; P) arbitrarily close to S u (f; ; P) for large i and the theorem readily follows. 5. The S-integral on spaces of maximal points We consider the S-integral of f, where f : X → R is a bounded function and X is homeomorphic to a subset of a continuous domain D that is dense in the Scott topology of D. We identify X with its homeomorphic image and henceforth assume that the embedding is an inclusion. Every domain P has a set of maximal points and these form a dense subspace. We henceforth assume that X lies in the set of maximal points. We refer the reader to [16,17,9], for further information about spaces of maximal points. We begin with the following standard lemma. Lemma 5.1. Suppose that f : X → R is bounded. We de ne a function fˆ : P → R, where X ↔ Max(P) ,→ P is a dense embedding, as follows: f̂(x) = sup inf {f(y): y ∈ (X ∩ ↑ z)}: z≪x ˆ ˆ = f(x) Then f(x)6f(x) for all x ∈ X and fˆ is Scott continuous. Furthermore, f(x) at each point of continuity of f. ˆ Proof. We note rst that all in ma in the de nition of f(x) exist, since ↑ z contains the non-empty Scott open set {w: z ≪ w}, and the latter must meet the dense set X . ˆ is well de ned, since f is a bounded function, and the supremum It is clear that f(x) ˆ and in mum exist uniquely. Now we claim that f(x)6f(x) for x ∈ X . Let x ∈ X and z ≪ x. Then we have x ∈ (X ∩ ↑ z), and thus inf {f(y): y ∈ X; y¿z}6f(x). Therefore ˆ f(x)6f(x). To show fˆ is Scott continuous, we need to show that fˆ is monotone and preserves the suprema of directed families. Let x ⊑ y where x; y ∈ P. Since z ≪ x ⊑ y implies that z ≪ y, we have {z: z ≪ x} ⊆ {z: z ≪ y}: ˆ ˆ Hence from the de nition f(x)6 f(y). Let {ui } be a directed family in P. Then we have immediately the following inequality: f̂ ↑ ui ¿ ↑ f̂(ui ); J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 271 ˆ since f is monotone. To show the other direction of the inequality, note that {z: z ≪ ui } ⊆ {z: z ≪ ui }; this is so because P is a continuous domain. Then we have the following: ↑ f̂ ui 6 ↑ sup {inf {f(y): y ∈ (X ∩ ↑ z)}}; z≪ui ˆ ↑ ui )6 ↑ f(u ˆ i ). Thus, fˆ is Scott continuous. i.e., f( ˆ To nish the proof we need only to show that for all ¿0, f(x) − 6f(x) for each point of continuity x ∈ X of f. Since f(x) is Scott continuous at x, there exists open subset U in X with x ∈ U such that f(U ) ⊆ (f(x) − ; ∞). Then there exists a Scott open subset W such that x ∈ W and W ∩ X ⊆ U . Hence, there exists z ∈ W such that z ≪ x. Thus, we have f̂(x) ¿ inf f(↑ z ∩ X ) ¿ inf f(W ∩ X ) ¿ inf f(U ) ¿ f(x) − : ˆ The following corollary follows easily from the de nition of f. Corollary 5.2. The extension fˆ de ned above is the largest Scott continuous extension of f if f is Scott continuous. Theorem 5.3. Let A be the crescent algebra generated by the open sets of a subspace X of a topological space Y and let  be a nitely additive measure on A. Then (B)  := (B ∩ X ) de nes a nitely additive measure on the crescent algebra of Y. Furthermore, a bounded continuous function f : Y → (R; Scott) satis es   f|X d: f d  = S S X Y Proof. A direct veri cation yields that  is de ned and nitely additive on the crescent algebra of Y . By Theorem 2.14 f is S-integrable with respect to  and f|X is S-integrable with respect to . For each ¿0, there exists a partition P contained in the crescent algebra of Y such that S u (f; ;  P) − S l (f; ;  P) ¡ : Let {B1 ; : : : ; Bn } denote those members of P that meet X non-trivially. Then since (B)  = 0 for other members of the partition, we have S u (f; ;  P) = n  sup f(Bi )(B  i) and S l (f; ;  P) = n  inf f(Bi )(B  i ): 1 1 Now P′ = {Bi ∩ X : i = 1; : : : ; n} is a partition of X by members of the crescent algebra of X , and n  1 sup f(Bi ∩ X )(Bi ∩ X ) 6 n  sup f(Bi )(B  i ): 1 − −  Thus, S u (f|X ; ; P′ )6S u (f; ;  P), and hence X f|X d6 Y f d .  Similarly −X f|X    d¿ −Y f d .  From this sandwiching we conclude that S Y f d  = S X f|X d. 272 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 Theorem 5.4. Let X ↔ Max(P) ,→ P be a dense embedding of X into a continuous domain P equipped with the Scott topology, let  be a nitely additive measure on the crescent algebra of X, and let f : X → (R; Scott) be a bounded function which is continuous a.e. Let  be de ned on the crescent algebra of P by (B)  = (B ∩ X ). ˆ = supz ≪ x inf {f(y): y ∈ (X ∩ ↑ z)}. Then Let fˆ : P → R be de ned by f(x)   S f̂ d :  f d = S P X Furthermore, if on the Scott open sets,  = valuations on P, then    f d = S f̂ di : f̂ d  = sup S S i P X ↑ i , a directed supremum of simple P Proof. By Lemma 5.1 the function fˆ is continuous into (R; Scott). Thus by the preceding theorem   f̂ d :  f̂|X d = S S P X ˆ X is equal to f a.e. Thus by Corollary 2.12 By Lemma 5.1 f|   f d f̂|X d = S S X and hence S X  X f d = S  P fˆ d .  The last assertion follows from Theorem 4.3. 6. The E-integral In this section, we review the de nition of the E(dalat)-integral introduced by Edalat and recently extended by Howroyd to arbitrary domains, recall some basic results concerning it, and establish its equivalence with our approach. As previously, let X ↔ Max(D) ,→ D be a dense embedding of X into the maximal points of a continuous domain D equipped with the Scott topology. Consider a bounded function f : X → R on X . Let  be a Borel probability measure on X such that (U  ) := (U ∩ X ) de nes a continuous valuation on the Scott open sets of D (this will be the case of the measure if  is continuous on the open sets if X or if the domain D is !-continuous). Since P 1 D is also a continuous domain with a basis of normalized simple valuations,  can be approximated by a chain of simple valuations on the domain D. Based on this idea, the E-integral can be introduced as follows.  De nition 6.1. Let  = b∈|| rb b ∈ P 1 E be a simple valuation, where || is the support of  and b is a point valuation for b ∈ E. Then the lower sum and upper sum of f with respect to  are de ned as  S l (f; ) = rb inf f(↑ b ∩ X ); b∈|| J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 273 and S u (f; ) =  rb sup f(↑ b ∩ X ); b∈|| respectively. The lower E-integral and upper E-integral of f with respect to  are de ned as  f d = sup{S l (f; ):  ≪ ;   simple}; E∗ and E-  ∗ f d = inf {S u (f; ):  ≪ ;   simple}; respectively. The bounded function f : X → R is said to be E-integrable with respect to  if  ∗  Ef d: f d = E∗  If f is E-integrable, the E-integral of f is denoted by E- f d and is de ned to be the value of the lower or upper integral:    ∗ f d: f d = EE- f d = E∗ ˆ Lemma 6.2. Let   f : X → R be bounded and let f be de ned as in Lemma 5.1. Then ˆ S D f d 6E f d. ∗ rb b ∈ P 1 D be a simple valuation such that  ≪ . Then from ˆ the de nition of fˆ (see Lemma 5.1), f(b)6 inf f(↑ b ∩ X ) for each b ∈ ||. It follows that    rb inf f(↑ b ∩ X ); rb f̂(b) 6 f̂ d = Proof. Let  = D  b∈|| b∈|| b∈||   and hence that D fˆ d6E- − f d. Since  is the directed supremum of all simple  ≪ , it follows from Theorem 4.3 that   S f d: f̂ d 6 ED ∗ Theorem 6.3. Let X ↔ Max(D) ,→ D be a dense embedding of X into the maximal points of a continuous domain D equipped with the Scott topology. Let  be a probability measure on X such that (U  ) := (U ∩ X ) de nes a continuous valuation 274 J.D. Lawson, B. Lu / Theoretical Computer Science 305 (2003) 259 – 275 on the Scott open sets of D. Let f : X → R be a bounded function. Then f is Edalat integrable if and only if f is continuous a.e., and in this case with  f is S-integrable  respect to the crescent algebra (or semialgebra) of X and S X f d = E- X f d. Proof. The assertion that f is Edalat integrable if and only if f is continuous a.e. is Theorem 12 of [11]. We know from  Theorem 5.4 that f is S-integrable with respect to the crescent algebra and S X f d = S D fˆ d ,  where fˆ is the Scott continuous extension of f to D. Furthermore, it follows from the previous lemma that   S D fˆ d6E- ∗ f d. Now we order dualize the whole argument for the dual Scott topology on R generated by all open lower rays (or, alternately, one can work with  −f). If we letf(x) be the upper semicontinuous extension of f to D, then we have  ∗ dually that E- f d6S D f d  = S X f d. The conclusion of the theorem now follows. We note that the only place in the proof that results of Howroyd are needed is in the assertion that a function that is Edalat integrable is continuous a.e. If one begins with a bounded function f : X → R that is continuous a.e., then our approach gives an alternate proof from Howroyd’s that the function is Edalat integrable (and yields in addition that this integral is equal to the S-integral). Indeed Howroyd’s results admit some extension. Consider the case that X ↔ Max(D) ,→ D is a dense embedding of X into the maximal points of a continuous domain D equipped with the Scott topology. Consider a bounded continuous function f : X → R on X . Let  be a bounded Borel measure on X such that (U  ) := (U ∩ X ) de nes a continuous valuation on the Scott open sets of D. If  is the directed supremum of simple valuations (no longer assumed normalized), then one can de ne the lower and upper sums of f with respect to these simple valuations. What goes wrong is that one can no longer trap the Edalat integral between the lower and upper sums (the latter may be too small), but one can ask whether they all converge to a unique value. And indeed they do, since one can trap them between the values of the S-integral for fˆ and f as in the preceding theorem, and the latter two must agree, since they must both equal S X f d. Thus one can approximate integrals using simple valuations for general bounded Borel measures, not only in the normalized case. One can declare that a bounded function f : X → R is continuous with respect to the algebra A is for every ¿0, there exists an A-partition P such that given x; y ∈ P ∈ P, |f(x)−f(y)|¡. It is a result of Rao and Rao (Chap. 4.7 of [3]) that if f is continuous with respect to A, then f is S-integrable with respect to all nitely additive measures on A. For X a topological space equipped with the crescent algebra, one observes that characteristic functions of open sets are continuous with respect to the crescent algebra, but are not continuous a.e. with respect to point measures in the boundary. Thus, the class of S-integrable functions includes a wider class of bounded functions than the Edalat integrable ones. However, from Theorem 2.15 we see that they are all Lebesgue integrable and the S-integral agrees with the Lebesgue integral. 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