THE GENERALIZED RIEMANN INTEGRAL ON LOCALLY COMPACT SPACES. Department of Computing. Imperial College. 180 Queen's Gate, London SW7 2BZ.

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1 THE GENEALIED IEMANN INTEGAL ON LOCALLY COMPACT SPACES Abbas Edalat Sara Negri Department of Computing Imperial College 180 Queen's Gate, London SW7 2B Abstract We extend the basic results on the theory of the generalized iemann integral to the setting of bounded or locally nite measures on locally compact second countable Hausdor spaces. The correspondence between Borel measures on and continuous valuations on the upper space U gives rise to a topological embedding between the space of locally nite measures and locally nite continuous valuations, both endowed with the Scott topology. We construct an approximating chain of simple valuations on the upper space of a locally compact space, whose least upper bound is the given locally nite measure. The generalized iemann integral is dened for bounded functions with respect to both bounded and locally nite measures. Also in this setting, generalized - integrability for a bounded function is proved to be equivalent to the condition that the set of its discontinuities has measure zero. Furthermore, if a bounded function is -integrable then it is also Lebesgue integrable and the two integrals coincide. Finally, we extend - integration to an open set and provide a sucient condition for the computability of the integral of a bounded almost everywhere continuous function. 1 Introduction Domain theory was introduced by Dana Scott in 1970 ([27]) as a foundational basis for the semantics of computation. In [6], a basic connection between domain theory and some main branches of mathematics has been established, giving rise, in particular, to a novel computational approach to measure theory and integration. A domain-theoretic framework for measure and integration has been provided in [7], by proving that any bounded Borel measure on a compact metric space can be obtained as the least upper bound of simple valuations (measures) on the upper space U, the set of non-empty compact subsets of ordered by reverse inclusion. Simple valuations approximating the given measure play the role of partitions in iemann integration and are used to obtain increasingly better approximations of the integral of a bounded real-valued function on a compact metric space. Thus, instead of approximating the function with simple functions as is done in Lebesgue theory, the measure is approximated with simple measures. This idea leads to a new notion of integration, called generalized iemann integration, -integration for short, similar in spirit but more general than iemann integration. All the basic results of the theory of iemann integration can be extended to this setting. For instance, it is proved in [7] that a bounded real-valued function on a compact metric space is -integrable with respect to a bounded Borel measure if and only if its 1

2 set of discontinuities has measure zero and that if the function is -integrable then it is also Lebesgue integrable and the two integrals coincide. This theory has had applications in exact computation of integrals [8], in the semantics of programming languages [12], in the 1-dimensional random elds Ising model in statistical physics [9], in forgetful neural networks [10], in stochastic processes [5] and in chaos theory [8]. Apart from the domain-theoretic integral, there are two other notions of generalized iemann integrals in the literature, namely, the McShane and the Henstock integrals [25, 15]. These are basically integrals for real valued functions on. Their generalizations to n also exist but they are much more involved. The basic McShane integral is equivalent to the Lebesgue integral with respect to to the Lebesgue measure. The Henstock integral, sometimes called the Henstock-Kurzweil integral, is a generalization of the McShane integral in the sense that any McShane integrable function is Henstock integrable but not conversely. The Henstock integral has the property that every continuous, nearly everywhere dierentiable function can be recovered by integration from its derivative. This property, which does not hold for the iemann or the Lebesgue integral, was historically the motivation behind the denition of this integral. The reason why the McShane and the Henstock integrals are called generalized iemann integrals is the following. In order to dene the ordinary iemann integral of a (bounded) function one partitions the domain of the function, whereas to dene the Lebesgue integral of a function one partitions its range to obtain an increasing sequence of simple functions converging pointwise to the given function. In the McShane integral, as well as in the Henstock integral, one returns to the idea of partitioning the domain of the function with a more sophisticated notion of \a tagged partition of an interval subordinate to a given positive valued function on the interval". Using such partitions one can obtain generalizations of the iemann integral and in fact that of the Lebesgue integral. The theory however is, like the Lebesgue theory, non-constructive and without any eective framework. The domain-theoretic generalization of the iemann integral works generally for integration of functions with respect to Borel measures on Polish spaces (topologically complete separable metrizable spaces) [13, 22, 17] which include locally compact second countable spaces. Here, one also deals with the domain of the function rather than its range. But now one goes beyond the notion of partitions and uses nite covers by open subsets to provide approximations to the measure. These approximations give generalized upper and lower sums with which we dene the integral. The theory, like the iemann theory, has a constructive and eective framework. The generalized iemann integral of a Holder function with respect to an eectively given measure can be approximated up to any desired accuracy by upper and lower sums [8]. In order to apply the generalized iemann integration to a wider range of problems, we look for an extension of the results in [7] and [8] to the more general setting of locally compact second countable Hausdor spaces with bounded or locally nite measures, the latter being measures which are nite on compact subsets of the space. A number of these extensions do require further work; others are straightforward generalizations of the compact case. These more general hypotheses, however, cover some central elds of application. For instance, probability distributions on the real line are examples of bounded measures on a locally compact space, whereas the Lebesgue measure on the real line is an example of a locally nite measure on a locally compact space. In this paper, after reviewing the domain-theoretic notions needed here, we show in section 3 that the set of locally nite measures on a locally compact space are in one to one correspondence with the set of locally nite continuous valuations on the upper space which are supported on its maximal elements. Indeed, this gives rise to a topological embedding when both spaces are endowed with the Scott topologies induced 2

3 by the pointwise order. In section 4, we show how to approximate a locally nite valuation on a locally compact space by means of an increasing sequence of simple valuations on the upper space, which is built up from any given presentation of the locally compact space as an increasing countable union of relatively compact open subsets. We proceed in section 5 with the denition of the generalized iemann integral of a bounded real-valued function on a locally compact space with respect to a bounded Borel measure. As in the case of compact spaces [7], generalized -integrability for a function f is seen to be equivalent to the condition that the set of discontinuities of f has measure zero. Moreover, if the -integral exists then the Lebesgue integral exists and the two integrals are equal. The notion of an eective approximation to a bounded measure by simple valuations on the upper space, which is used to compute integrals of Holder continuous functions up to any precision, has a straightforward extension to locally compact spaces. We then dene, in section 6, the -integral with respect to a locally nite measure. In the case of functions with compact support, the -integral reduces to the -integral in compact spaces with respect to the restriction of the measure to the support of the function. In the case of functions with non-compact support, the -integral is dened in a way similar to the improper iemann integral. It is shown, in this more general case, that if a function is -integrable then it is also Lebesgue integrable and the two integrals coincide. Finally in section 7, following the work in [4], we extend the denition of - integration to an open set, and prove that the lower integral coincides with the Lebesgue integral on the open set, whereas the upper integral coincides with the Lebesgue integral on its closure. As a consequence, when the boundary of the open set has measure zero, we obtain two sequences that converge from below and from above to the value of the integral of a bounded almost everywhere continuous function. This allows a computation of the integral on the open set up to any desired degree of accuracy. 2 Valuations on continuous posets We rst recall some basic notions from domain theory which we need in this paper. A non-empty subset A P of a partially ordered set (poset) (P; v) is directed if for any pair of elements x; y 2 A there is z 2 A with x; y v z. A directed complete partial order (dcpo) is a partial order in which every directed subset A has a least upper bound (lub), denoted by F A. An open set O P of the Scott topology of P is a set which is upward closed (i.e. x 2 O & x v y ) y 2 O) and is inaccessible by lubs of directed sets (i.e. if A is directed with a lub, then F A 2 O ) 9x 2 A: x 2 O). The Scott topology of any poset is T 0. Given two elements x; y in a poset P, we say x is way-below y or x approximates y, denoted by x y, if whenever y v F A for a directed set A with lub, then there is a 2 A with x v a. We sometimes write P to emphasise that the way-below relation is with respect to P. We say that a subset B D is a basis for D if for each d 2 D the set A of elements of B way-below d is directed and d = F A. We say D is continuous if it has a basis; it is!-continuous if it has a countable basis. In any continuous poset, subsets of the form "b = fx j b xg where b belongs to a basis give a basis of the Scott topology. A valuation on a topological space Y is a measure-like function dened on the lattice (Y ) of open sets. Here is the precise denition. Denition 2.1 A valuation on a topological space Y is a map which, for all U; V 2 (Y ), satises : (Y )! [0; 1) 3

4 1. (U) + (V ) = (U [ V ) + (U \ V ); 2. (;) = 0; 3. U V ) (U) (V ). A continuous valuation is a valuation such that whenever A is a directed family in (Y ) then 4. ( S O2A O) = sup O2A (O). A locally nite continuous valuation is a map : (Y )! [0; 1] satisfying the same properties of a continuous valuation and 5. (V ) < 1 for V Y. The set of locally nite continuous valuations on Y is denoted by P `(Y ); it is pointwise ordered by: 1 v 2 (8O 2 ())( 1 (O) 2 (O)) : The subposet of continuous valuations bounded by one, i.e. with (Y ) 1, is denoted by P (Y ) and the set of normalized valuations, i.e. those with (Y ) = 1 is denoted by P 1 (Y ). The point valuation based at b 2 Y is the valuation dened by b : (Y )! [0; 1) b (O) = 1 if b 2 O 0 otherwise. Any nite linear combination n i=1 r i bi (r i 2 [0; 1); i = 1; : : :; n) of point valuations is called a simple valuation. If Y is an (!)-continuous dcpo then P (Y ) is also an (!)-continuous dcpo with a basis of simple valuations [19]. We have the following property: Proposition 2.2 [20, page 46] Let = P b2b r b b be a simple valuation and a continuous valuation on a continuous dcpo Y. Then i for all A B we have b2a where "A = fy 2 Y j 9a 2 A: a yg. r b < ("A); 2 If Y is an (!)-continuous dcpo with bottom then P 1 (Y ) is also an (!)-continuous dcpo with a basis of simple valuations and we have the following results [7]: Lemma 2.3 Let Y be a dcpo with bottom and let 1 = r a a 2 = s b b a2a b2b be simple valuations in P 1 (Y ). Then we have 1 v 2 i, for all a 2 A and for all b 2 B, there exists t a;b 0 such that P b2b t a;b = r a, P a2a t a;b = s b and t a;b > 0 implies a v b. 4

5 Dene the maps m + : P Y! P 1 (Y ) m : P Y! P Y 7! + 7! where and (O) if O 6= Y + (O) = 1 otherwise (O) if O 6= Y (O) = (Y f?g) otherwise. Proposition 2.4 (i) P Y & 2 P 1 Y ) + P 1 Y. (ii) P 1 Y ) P Y v. Lemma 2.5 Suppose ; 2 P 1 Y. Then P 1 Y implies (Y f?g) < 1. Proposition 2.6 For two simple valuations 1 = r b b 2 = s c c b2b c2c in P 1 Y, we have 1 P 1 Y 2 i? 2 B with r? 6= 0, and, for all b 2 B and all c 2 C, there exists a nonnegative number t b;c with t?;c 6= 0 such that and t b;c 6= 0 implies b c. c2c t b;c = r b b2b t b;c = s c When considering locally nite valuations closure under directed joins does not hold any more, i.e. P `(Y ) is not a dcpo. However, it is still true that locally nite valuations can be approximated by simple valuations, since this results holds in general for (unbounded) valuations (cf. theorem 5.2, page 46 in [20]), and simple valuations are in particular locally nite. So we have: Proposition 2.7 For any continuous dcpo Y, any 2 P `(Y ) is the supremum of the (directed) set of simple valuations way-below it and, therefore, P `(Y ) is a continuous poset. 3 Locally nite measures on a locally compact space Throughout the paper, will denote a second countable locally compact Hausdor space. We will use the decomposition = i2in i, where h i i i2in is an increasing sequence of relatively compact open subsets of such that i i+1. We start with some denitions: Denition 3.1 A Borel measure on a locally compact Hausdor space is locally nite if (C) < 1 for all compact C. M `() will denote the set of locally nite measures on. The set of measures bounded by one and the set of normalized measures are denoted respectively by M() and M 1 (). 5

6 We recall from [6] that the upper space U of a topological space is the set of all non-empty compact subsets of, with the base of the upper topology given by the sets 2a = fc 2 U j C ag; where a 2 (). When is a second countable locally compact Hausdor space, then the upper space U of is an!-continuous dcpo and the Scott topology of (U; ) coincides with the upper topology. The lub of a directed subset is the intersection and A B i B is contained the interior of A. The singleton map s :! U with s(x) = fxg is a topological embedding onto the set of maximal elements of U. In [6], it was shown that the map M()! P (U) 7! s 1 is an injection into the set of maximal elements of P (U) and it was conjectured that its image is the set of maximal elements. This conjecture was later proved by Lawson in a more general setting [23]. The continuous dcpo U does not necessarily have a bottom element. Therefore, in order to consider normalized valuations, we will adjoin a bottom element?= and denote the dcpo with bottom thus obtained with (U)?. Then the injective map 7! s 1 : M 1 ()! P 1 (U)? is onto the set of maximal elements of P 1 (U)?. Here, we will show a one-to-one correspondence between locally nite Borel measures on and locally nite continuous valuations on the upper space supported in s(). Proposition 3.2 Let s :! U be the singleton map. Then the map is well dened. e : M `()! P `(U) 7! s 1 Before proving the above proposition we need the following lemmas, connecting the way-below relation on the upper space U of with the one on. Lemma 3.3 Let fo i : i 2 Ig be a directed family in (). Then [ [ 2( O i ) = 2O i : i2i i2i Proof: The inclusion from right to left trivially holds. For the converse, assume that C is a non-empty compact subset of S i2i O i. By compactness, C has a nite subcover, and therefore, since the family of opens is directed, there exists i 2 I such that C is a subset of O i. 2 Lemma 3.4 Let V be an open set in the Scott topology of U. Then V U in (U) if and only if s 1 (V ) in (): Proof: ): Suppose S i2i O i where the right-hand side is a directed union and O i 2 (). Then, by lemma 3.3, we have U = 2 S i2i 2O i. Since V U in (U), there exists i 2 I such that V 2O i, and therefore s 1 (V ) O i, thus proving the claim. (: Suppose U S i2i O i, where the right-hand side is the union of a directed family of opens of the upper space. Since by hypothesis s 1 (V ) and is a locally compact space, there exists a compact set C such that s 1 (V ) C. Since C 2 U, 6

7 there exists i 2 I such that C 2 O i and therefore " C O i as O i is open in the Scott topology of U and thus an upper set with respect to reverse inclusion. This implies V " C. For, if K 2 V and x 2 K, then K v fxg and, hence, fxg 2 V since V is upward closed. Thus x 2 s 1 (V ) C, i.e. K C. Therefore V " C O i, i.e., V U. 2 Proof of proposition 3.2: Let be a locally nite Borel measure on. Then it is immediate to verify that s 1 satises conditions 1-4 of denition 2.1 since is a measure and s 1 preserves (directed) unions and intersections. The continuous valuation s 1 is locally nite since, if V U, then, by lemma 3.4, s 1 (V ). Since is locally compact, there exists a compact subset K of such that s 1 (V ) K. Therefore, by monotonicity of, s 1 (V ) (K), and by the assumption on local niteness of the conclusion follows. 2 By a result from [24] (theorem 3.9), if P is a continuous dcpo equipped with its Scott topology, then every continuous valuation on P extends uniquely to a measure, denoted by. Thus, in particular, s 1 extends to a unique measure (s 1 ) on U, which, by abuse of notation, we will denote by too. Such a measure satises (U s()) = 0, that is we have: Proposition 3.5 The valuation s 1 is supported in s(). Proof: We have U s() = S i2in (U i s( i )). By proposition 5.9 and corollary 5.10 in [6], s()ands( i ) are G subsets of U and there exists a countable decreasing sequence of opens ho i;j i j2in such that s( i ) = \ j2in O i;j and s 1 (O i;j ) = i. Therefore (U s()) = sup (U i s( i )); i2in so it is enough to prove that, for all i 2 IN, (U i s( i )) = 0. We have (s( i )) = inf j2in (O i;j ) = inf j2in s 1 (O i;j ) = ( i ) = s 1 (U i ) = (U i ). It follows that (U i s( i )) = 0. 2 Adapting the notation from [6], we will denote with S`(), resp. S 1 (), the locally nite valuations, resp. the normalized valuations, on U which are supported in s(). We have: Theorem 3.6 The map is a bijection with inverse given by e : M `()! S`() 7! s 1 j : S`()! M `() 7! s : The proof of the theorem is based on the following lemma. Lemma 3.7 If and are locally nite Borel measures which have the same restriction to () then =. Proof: We have ( i ) = ( i ) < 1. Furthermore, for all Borel sets B, B \ i is a Borel set in the induced topology of i. By proposition 5.2 in [6], any nite continuous valuation on a locally compact Hausdor space has a unique extension to a measure, so that we have ( i \B) = ( i \B) and therefore (B) = ( \B) = (( S i2in i)\b) = ( S i2in ( i \ B)) = sup i2in ( i \ B) = sup i2in ( i \ B) = (B). 2 Proof of theorem 3.6: By the foregoing discussion, e is well dened. To prove that j is well dened we only have to prove j() is a locally nite measure when is a locally nite continuous valuation. Let C be a compact subset of. There exists a relatively compact 7

8 open subset O, i.e. O, with C O. Then (j())(c) (j())(o) = (s(o)) (2O) < 1, where the last inequality holds since 2O U by lemma 3.4. The proof that j is the inverse of proceeds as in [6], theorem In [8] it has been proved that the space of probability measures of a compact metric space equipped with the weak topology is topologically embedded, via the map 7! s 1, onto the subspace of the maximal elements of the probabilistic power domain of the upper space. Here a topological embedding can be obtained by considering the Scott topologies on M `() and P `(U): We dene a partial order on M `(), as in P `(U), by 1 v 2 (8O 2 ())( 1 (O) 2 (O)) : The relation v is clearly reexive and transitive, and it is antisymmetric by lemma 3.7. Observe that the space (M `(); v) is not a dcpo. In fact, a directed subset D of M `() has least upper bound in M `() if and only if for all open sets O such that O in () the set f(o) : 2 Dg is bounded. It is immediate to check that the maps e and j are continuous with respect to the Scott topologies on M `() and P `(U), i.e., e is a topological embedding. 4 Approximation of locally nite measures Given a measure on a compact metric space it is possible to construct a chain of simple valuations on the upper space with the measure as its least upper bound (cf. [8]). In this section we will generalize this construction to the case of locally nite measures on a locally compact space. The relatively compact subsets i are used to construct a sequence of simple valuations on U approximating any given locally nite measure on, thus generalizing the procedure which is worked out in [8]. As in that work, we assume that the measure is given by its values on a given countable basis closed under nite unions and nite intersections. Since the subsets i are relatively compact, for all i 2 IN there exists an ordered nite covering C i of i made of relatively compact open sets of the basis with diameter 1 i. Moreover such coverings can be chosen satisfying the additional requirement [C i [C i+1. These hypotheses are needed in order to obtain an increasing sequence of simple valuations. We assume in this section, and the rest of the paper, that open covers are always constructed from the given countable base satisfying the above property. For each i 2 IN we dene inductively a nite ordered open cover D i for i as a list of opens sets in the following way, where the symbol denotes the concatenation operation for lists: D 1 C 1 ; D 2 D 1 ^ C 2 ho 2 C 2 : O = [ C 1 i; D 3 D 2 ^ C 3 ho 2 C 3 : O = [ C 2 i; : : : D i+1 D i ^ C i+1 ho 2 C i+1 : O = [ C i i; : : : where, for given ordered covers A and B, A ^ B denotes the cover ho 1 \ O 2 : O 1 2 A; O 2 2 Bi ordered lexicographically. We recall from [8] that, for an ordered cover A = ho 1 ; : : :; O n i, the simple valuation A associated with A is given by A n i=1 r i Oi, where r i = (O i [ j<i O j ): 8

9 We will denote with i the simple valuation Di associated with D i. We will now prove that the i 's constitute a chain of simple valuations with least upper bound. The following lemma is proved as proposition 3.1 in [8]. Lemma 4.1 Let A = ho 1 ; : : :; O n i be an ordered cover of i with O j for all j n, a locally nite measure on and the r i 's as above. Then for all open subsets O of i we have r i (O) r i : O io O i\o6=; Corollary 4.2 For all i 2 IN, i v e() in P `(U). Proof: Let V = S 2 2O be an open set of the upper space. Then i (V ) = P O i2 S 2 2O r i P O i S 2 O r i ( S 2 O ) = e()(v ) : where the last inequality holds by lemma Proposition 4.3 For all i 2 IN, i v i+1 in P `(U). Proof: Let O 2 (U), D S i = ho 1 ; : : :; O n i, C i+1 = hv 1 ; : : :; V m i and let, for 1 i n, 1 j m, r i = (O i P S i 0 <i O i 0), r (i;j) = (O i \ V j (i 0 ;j 0 O )<(i;j) i 0 \ V j0). Then i+1 (O) Di^C i+1 (O) = O r i\v j2o (i;j). Furthermore, r (i;j) = r (i;j) : P O i2o P O i\v j2o r (i;j) jm O i2o O i2o jm As in the proof of proposition 3.4 in [8], it follows that P jm r (i;j) = r i, and therefore jm r (i;j) P O i2o r i = i (O), and the result follows. 2 Proposition 4.4 = F i2in i. Proof: Since, for all i, i v, we have F i2in i v. For the converse inequality, it is sucient by lemma 3.3 in [8] to check that, for all opens O 2 (), sup i i (2O) (O). We distinguish two cases, i.e. (O) < 1 and (O) = 1. If (O) < 1, let O j O \ j, for j 2 IN. Then, for all j, O j and S O = O j2in j. By F theorem 3.5 in F [8], we F have, for all F F j 2 IN, F(O j ) = F i2in i(o j ) and therefore (O) = (O j2in j) = j2in i2in i(o j ) = i2in j2in i(o j ) = i2in i(o). If (O) = 1, then for all M > 0 there exists V O such that (V ) > M. Since V O and is F a locally nite measure, (V ) is nite and F therefore, F by the previous case, (V ) = i (2V ). Thus, since 2O 2V, we have i (2O) i (2V ) = (V ) > M, and the conclusion follows since M is arbitrary. 2 5 Integration with respect to bounded measures Let be a Borel measure on such that 0 < () < 1. By rescaling, we can suppose that is normalized, i.e. that () = 1. By corollary 3.3 in [7], P 1 (U)?, the set of normalized valuations of the probabilistic power domain of (U)?, is an!-continuous dcpo with a basis of normalized simple valuations. 9

10 We also know from the previous section that there exists an increasing chain of simple valuations h i i i0 in P (U)? such that s 1 = sup i0 i. For convenience, in the rest of this section, we identify s 1 with and, therefore, write = sup i0 i. The valuations + i i + (1 i ((U)? ))? are in P 1 (U)?. Moreover, for any proper open subset O of (U)?, we have + i (O) = i (O) and s 1 = sup i0 + i, where h+ i i i0 is again an increasing chain of simple valuations. Let f be a bounded real-valued function dened on and let be a bounded Borel measure on. For a simple valuation P b2b r b b, dene, as in [7], the lower and upper sums by S`(f; ) b2b r b inf f[b] S u (f; ) b2b r b sup f[b]; where f[b] = ff(x)jx 2 bg. Observe that, since f is bounded, S` (f; ) and Su (f; ) are well dened real P numbers. For a choice of x b 2 b for all b 2 B, we also have a iemann sum S x (f; ) = b2b r bf(x b ). By proposition 4.2 in [7] we have: Proposition 5.1 Let 1 ; 2 2 P 1 (U)? be simple valuations with 1 v 2. Then S` (f; 1) S` (f; 2) and S u (f; 2) S u (f; 1). Corollary 5.2 If 1 ; 2 2 P 1 (U)? are simple valuations with 1 ; 2, then S` (f; 1) S u (f; 2) Using the notation introduced above we dene the lower and upper -integrals as follows: Denition 5.3 fd sup S` (f; ) (the lower -integral); fd inf S u (f; ) (the upper -integral). By corollary 5.2 we have fd fd : Denition 5.4 We say that f is -integrable with respect to and write f 2 () if f = f. As a consequence of the denition we have: Proposition 5.5 (The -condition). We have f 2 () i for all > 0 there exists a simple valuation 2 P 1 (U)? with such that S u (f; ) S` (f; ) <. If f is -integrable then the integral of f can be calculated by using the increasing sequence of simple valuations h + i i i0 : Proposition 5.6 If f is -integrable with respect to, and h i i i0 sequence of simple valuations on U with least upper bound, then fd = sup S`(f; + i ) = inf S(f; u + i ): i0 i0 is an increasing 10

11 Proof: Cf. proposition 4.9 in [7]. 2 The following properties are easily shown as in [7]: Proposition If f and g are -integrable with respect to then f + g is also -integrable with respect to and (f + g)d = fd + gd. 2. If f is -integrable with respect to and c 2 I then cf is -integrable with respect to and cfd = c fd. 3. If f and g are -integrable with respect to then so is their product fg. 5.1 Lebesgue criterion for -integrability We will extend the Lebesgue criterion for iemann integration of a bounded function on a compact real interval to -integration of a bounded real-valued function on a locally compact space, by generalizing the result in [7], that is the analogous criterion for - integration on compact spaces. We rst recall some denitions. Denition 5.8 Let T and r > 0 and dene f (T ) supff(x) f(y) j x; y 2 T g, called the oscillation of f on T ;! f (x) lim h!0 + f (B(x; h)), where B(x; h) is the open ball of radius h centred at x; D r fx 2 j! f (x) 1 r g. Then we have (cf. [2], p. 170, and [7]): Proposition 5.9 The following statements hold: (i) f is continuous at x i! f (x) = 0. (ii) If is compact and! f (x) < for all x 2, then there exists > 0 such that, for all compact subsets b with jbj <, we have f (b) <. (iii) For any r > 0, the set D r is closed. If D is the set of discontinuities of f, we have D = S n1 D n where D 1 D 2 D 3 : : : is an increasing chain of closed sets. Hence D is an F and therefore a Borel set. In the following, 2 M 1 (). We have: Lemma 5.10 Let d be a compact subset of and let = P b2b r b b 2 P 1 (U). Then: 1. If v then P b\d6=; r b (d). 2. If then P b \d6=; r b (d), where b denotes the interior of b. Proof: Cf. lemma 6.4 in [7]. 2 Proposition 5.11 Let h i i i2i be a directed set of simple valuations i = b2b i r i;b b in P 1 (U). Then F i2i i 2 im(e) i all > 0 and all > 0, there exists i 2 I with b2b i;jbj r i;b < where jbj is the diameter of the compact set b. 11

12 Proof: The two directions in the proof were shown for a compact in proposition 4.14 in [7] and in proposition 5.1 in [8], but the proofs there hold for a locally compact as well. 2 We recall from [26] that a bounded measure on a second countable locally compact Hausdor space is regular, i.e., for any Borel set B, it satises (B) = inff(o) j B O; O openg = supf(k) j K B; K compactg : Then the Lebesgue criterion follows: Theorem 5.12 A bounded real-valued function on a locally compact second countable Hausdor space is -integrable with respect to a bounded Borel measure on i its set of discontinuities has measure zero. Proof: Suppose (D) > 0. Since D = S n1 D n, there exists n 1 such that (D n ) > 0. As D n = S i0 D n \ i, there exists i such that (D n \ i ) > 0. We put D 0 n D n \ i. Let = P b2b r b b be a simple valuation with. Then S u (f; ) S`(f; ) = P b2b r b(sup f[b] inf f[b]) P b \D 0 n 6=; r b(sup f[b] inf f[b]) P b \D 0 n6=; r b=n (D 0 n)=n > 0 by denition of D 0 n and the above lemma. Thus f does not satisfy the -condition and therefore it is not -integrable. Conversely, assume (D) = 0. For all n 1, there exists a compact subset Y of such that ( Y ) < 1 n. Let T and be regular compact subsets of with Y and T. Let D n = fx 2 T j w f (x) 1 n g. We know that D n is closed and (D n ) = 0. By regularity of, there exists an open V such that D n \ V and (V ) < 1 n. By taking the intersection of V with T, if necessary, we can assume V T. Since D n \ is compact, there exists an open set W with D n \ W and W V. Let d(w ; V c ) = 1 and = 2, denotes the boundary of. Observe that W is compact and for all x 2 W, w f (x) < 1 n. By proposition 5.9(ii), there exists 3 such that for all compact b W with jbj < 3 we have f (b) < 1 n. Let 0 < < min( 1; 2 ; 3 ). By proposition 5.11, there exists 2 P 1 (U)?, = P b2b r b b, such that P jbj r b < 1 n and v. Observe that, by the choice of, jbj < implies b or b Y. Moreover, jbj < and b imply b V \ or b W. Then we have S u (f; ) S`(f; ) = b2b r b (sup f[b] inf f[b]) r b (sup f[b] inf f[b]) + jbj + jbj<;b Y jbj<;b r b (sup f[b] inf f[b]) r b (sup f[b] inf f[b]) + jbj + jbj<;b W jbj<;bv \ r b (sup f[b] inf f[b]) + r b (sup f[b] inf f[b]) r b (sup f[b] inf f[b]) jbj<;b Y r b (sup f[b] inf f[b]) (M m)=n + (M m)=n + 1=n + (M m)=n = [3(M m) + 1]=n and we conclude that f is -integrable. 2 12

13 5.2 -integration and Lebesgue integration In this section we will prove that the result in [7] connecting -integration and Lebesgue integration for bounded real-valued functions dened on a compact space extends to the case of bounded functions dened on a locally compact second countable Hausdor space. The proof for the compact case uses the domain-theoretic result for the existence of a directed set of deations whose lub is the identity function on the domain; it can be extended to the locally compact case using the one-point compactication of the space. However, the following alternative proof which uses the construction of the simple valuations in section 4 approximating a given measure is elementary and conceptually simpler than the proof in [7]. Furthermore, the new proof can be used to generalize the result to other spaces such as complete separable metric spaces where the domain-theoretic property on the existence of deations does not exist. Theorem 5.13 If a bounded real-valued function f is -integrable with respect to a bounded Borel measure on a locally compact second countable Hausdor space, then it is also Lebesgue integrable and the two integrals coincide. Proof: We will use the F increasing chain of simple valuations h i i i2in, constructed in section 4, such that = i2in i. We recall that the simple valuation i associated with the ordered cover D i = ho i;1 ; : : :; O i;ni i of i is given by i = n i j=1 r i;j Oi;j, where r i;j = (O i;j [ k<j O i;k ) : Let V i;j O i;j S k<j O i;k. Observe that for all x 2 [D i there exists a unique j such that x 2 V i;j. Moreover, let m and M be a lower bound and an upper bound of f on, respectively. For all i 2 N we dene two functions f i : [D i! I f + i : [D i! I x 7! inf f[v i;j ]; for x 2 V i;j x 7! sup f[v i;j ]; for x 2 V i;j By putting f i (x) = m and f + i (x) = M for x 2 [D i we obtain two functions f i :! I f + i :! I Observe that f i and f + i are simple measurable functions. Moreover, since the cover D i+1 is a renement of the cover D i, for all x 2 we have Let m : : : f i (x) f i+1 (x) : : : f(x) : : : f + i+1 (x) f + i (x) M : f :! I f + :! I x 7! lim i!1 fi (x) x 7! lim i!1 f i + (x): Then f (x) f(x) f + (x) for all x 2, and, by the monotone convergence theorem, f and f + are Lebesgue integrable. We will now estimate their Lebesgue integrals. We have S`(f; i ) = P n i j=1 (V i;j) inf f[o i;j ] and P n i j=1 (V i;j) inf f[v i;j ] = L f i d m ( [D i) S u (f; i ) = P n i j=1 (V i;j) sup f[o i;j ] P n i j=1 (V i;j) sup f[v i;j ] = L f + i d M ( [D i) 13

14 Since f i f i + implies L f i d L f i + m ( [D i ) + S`(f; i ) L d, we obtain f i d L f i + d Su (f; i ) + M ( [D i ): Since f is assumed to be -integrable, we know by proposition 5.6 that S`(f; i ) increases to fd and S u (f; i ) decreases to fd. Moreover ( [D i )! 0 as i! 1 since S i>0 [D i =. Therefore, both L f i d and L f + i d converge to fd, and thus, by the monotone convergence theorem, L f d = L f + d = fd. We thus obtain L (f + f )d = 0, which implies that f + = f almost everywhere. Therefore f = f + = f almost everywhere and we can conclude that f is Lebesgue integrable and that L fd = L f d = L f + d = fd as required Computation of integrals Following [8], we can develop an eective framework for computing integrals of bounded Holder continuous functions with respect to a normalized measure on a locally compact second countable metric space. This is a straightforward generalization of the compact case and is presented here for completeness. Given a measure 2 M 1 (), a chain h i i i2n of simple valuations in P 1 (U) is an eective approximation to if F i2i i = and for all positive integers m and n there exists i 0, recursively given in terms of m and n, such that i = P c2c r c c satises P jcj1=m r c < 1=n. Suppose such an eective approximation exists for. Let f :! be a bounded Holder continuous function with constants k 0 and h > 0 such that jf(x) f(y)j k(d(x; y)) h for all x; y 2, and let jf(x)j K for all x 2. In this setting one can compute the integral of f with respect to up to any desired accuracy as follows. Let > 0 be given. To compute f d to within accuracy, we choose positive integers m and n with 1=m < (=2k) 1=h and 1=n < =(4K), and let the integer i be such that i = P c2c r c c satises P jcj1=m r c < 1=n. We have S`(f; i ) f d S u (f; i ); S`(f; i ) S (f; i ) S u (f; i ); where S (f; i ) is any generalised iemann sum for i. For any c 2 C we have sup f[c] inf f[c] 2K; whereas for c 2 C with jcj < 1=m we have sup f[c] inf f[c] < =2. Hence, j f d S (f; i )j S u (f; i ) S`(f; i ) = = jcj1=m =2 + =2 = : N c2c r c (sup f[c] inf f[c]) + r c (sup f[c] inf f[c]) jcj<1=m r c (sup f[c] inf f[c]) Therefore any iemann sum for i gives the value of the integral up to accuracy. We have then shown: Theorem 5.14 The expected value of any Holder continuous function with respect to any normalised measure on a second countable locally compact metric space can be obtained up to any given accuracy with an eective approximation of the measure by an increasing chain of normalised valuations on the upper space of the metric space. 14

15 6 Integration with respect to locally nite measures 6.1 Functions with compact support In this section we dene the -integral for a real-valued function with compact support dened on a locally compact space with respect to a locally nite measure. We rst need the following: Denition 6.1 For any Borel measure and any Borel set B, let j B be the unique extension to a measure of the continuous valuation given by j B (O) (B \ O) : Let f be a real-valued function on with compact support C and let be a locally nite measure on. Then the integral of f with respect to is given by fd = fdj C : The computation of the above integral can be obtained by cutting down, to the compact support C of f, the construction of the chain of simple valuations approximating the given locally nite measure. F ecall from section 4 that = P i, where i is the simple valuation, S associated with n the ordered cover D i of i, given by i r j=1 i;j Oi;j, with r i;j = (O i;j k<j O i;k). Since C is compact, there exists n 2 IN such that C i for all i > n. For i > n, dene C i = n i j=1 r C i;j Oi;j\C, where rc i;j = (O i;j \ C [ k<j O i;k \ C) : The following properties are then easily derived as in section 4 (observe that it is enough to know the value of on the induced topology of C): Lemma 6.2 P O i;j\co rc i;j (O \ C) P O i;j\c\o6=; rc i;j : Then we have: Proposition 6.3 Let j C and C i 1. For all i > n, C i v j C ; 2. For all F i > n, C i v C ; i+1 3. j C = i>n C i. be dened as above. Then: The above proposition shows that the integral of f can be computed by means of the chain of simple valuations h C i i i. Therefore -integrability of f with respect to reduces to the -integrability of fj C with respect to j C. 6.2 Functions with non-compact support Let f :! I be a positive function which is bounded on compact sets. Given a subset A of let A denote the characteristic function of A. In order to extend the theory of generalized iemann integration to this setting we need the following. Denition 6.4 A positive function f, bounded on compact sets, is -integrable on with respect to if, for all i 2 IN, the function with compact support f i is -integrable and the limit lim f i d i!1 15

16 is nite. In this case we put fd = lim f i d : i!1 The above denition of -integrability for positive functions bounded on compact sets does not depend on the S choice of the increasing sequence of relatively compact subsets i 's such that = i i. Indeed, if hy i i i2in is another such sequence then, for all i, f i is -integrable if and only if, for all i, f Yi is -integrable and lim f i d = lim i!1 i!1 In fact, for all i, there exists j such that i Y j, hence f i d f Yi d: f Yi d and conversely. For a general function f we can use the decomposition f = f + f where f + = max(f; 0) and f = max( f; 0), and say that f is -integrable if both f + and f are -integrable and we put fd = f + d f d : The following result connects -integration with respect to locally nite measures to Lebesgue integration, generalizing theorem Theorem 6.5 If a real-valued function f bounded on compact sets is - integrable with respect to a locally nite Borel measure on a locally compact second countable Hausdor space, then it is also Lebesgue integrable and the two integrals coincide. Proof: By using the decomposition f = f + f and the fact that Lebesgue integrable functions are closed under sum, we can suppose f 0. By hypothesis, for all i, f i is -integrable. Since it is a function with compact support, by the results of the previous section, its -integral can be computed with respect to the bounded measure j i. By theorem 5.13, it is Lebesgue integrable and L f i d = f i d : Moreover hf i i i2in is an increasing monotonic sequence of functions that converges pointwise to f. Therefore, by the monotone convergence theorem, f is Lebesgue integrable and L fd = lim L f i d : i!1 Since lim i!1 L f i d = lim i!1 f i d = fd, we have L fd = fd. 2 16

17 7 The generalized iemann integral on an open Set In this section we generalize the denition of -integration to an open set, in such a way that it gives the usual one when the open set is the whole space. We remark that for this purpose we could take O itself as a locally compact space and then use the results already established in the theory of integration developed so far. Nevertheless this extension is necessary if we want to compute the integral of f on O by using the original chain of simple valuations h i i i2in approximating on. Once this chain has been obtained, it will not be necessary to construct a new chain for each subspace O. In what follows, f is a bounded non-negative real valued function on and a normalized Borel measure on. P Denition 7.1 Let = a2a r a a be a simple valuation. upper Darboux sums relative to the open O are, respectively: The generalized lower and SÒ(f; ) P a2a;ao r a inf f[a] ; S u O (f; ) P a2a;a\o6=; r a sup f[a] : Note that for O =, the above sums reduce to the earlier denitions. As in section 5, we will consider the!-continuous dcpo with bottom (U)? and obtain the generalization of proposition 5.1 for the lower and upper sums relative to an open set. Proposition 7.2 Let 1 ; 2 2 P 1 (U)? be simple valuations with 1 v 2. Then SÒ(f; 1) SÒ(f; 2) and S u O (f; 2) S u O (f; 1). Proof: The rst inequality holds as before. For the second, by the above lemma, we have: S u O(f; 2 ) = = b2b;b\o6=; b2b;b\o6=; a2a;a\o6=; a2a;a\o6=; s b sup f[b] = t a;b sup f[b] r a sup f[a] S u O(f; 1 ) : 2 b2b;b\o6=; a2a t a;b sup f[b] b2b;b\o6=; a2a;a\o6=; t a;b sup f[a] Denition 7.3 The lower -integral of f on O with respect to is fd = sup O SÒ(f; ). The upper -integral of f on O with respect to is O fd = inf SO u (f; ). We clearly have fd fd = sup O O SÒ(f; ). Proposition 7.4 If f is a non-negative, real-valued function on which is P continuous almost everywhere with respect to, then for all > 0 there exists, = b2b r b b, such that r b sup f[b] SÒ(f; ) < : b2b;bo Proof: The proof proceeds as the proof of the `if' part of theorem 5.12 by showing P P that for all > 0 there exists a simple valuation = b2b r b b, such that b2b;bo r b(sup f[b] inf f[b]) <. 2 17

18 7.1 -integration on an open set and Lebesgue integration We will now investigate the connection between -integration on an open set and Lebesgue integration. We will see that when the function f is continuous almost everywhere and f 0 then the lower integral coincides with the Lebesgue integral of f on O whereas the upper integral coincides with the Lebesgue integral of f on O. An important consequence of this fact is that the lower and the upper integral are equal when the boundary of O has measure zero. In what follows, is a normalized Borel measure on, f is a bounded, non-negative, real-valued function on whose set of discontinuities has measure zero and O is an open subset of. F Lemma 7.5 Let = 2D be the lub of a directed set of simple valuations on U and O an open set. For F P P = b2b r b b, dene O = b2b r ;bo b b. Then the set f O : 2 Dg is directed and 2D O = j O. Proof: First we prove that for two simple valuations and, v implies O v O and therefore f O : 2 Dg is directed: For any open A U we have O (A) = (A \ 2O) 0 (A \ 2O) = 0 O (A). Then we have j O (A) j O s 1 (A) (O \ s 1 (A)) = j O s 1 (2O \ A) F = (2O \ A) = ( 2D )(2O \ A) = F 2D O(A) :2 Lemma 7.6 If f is a non-negative real valued function on which is continuous almost everywhere with respect to then fd = supf O C fd j C O; C compactg : Proof: For O =, the result is clear. Assume that O 6=. We rst prove sup SÒ(f; ) supf fd j C O; C compactg: C P S, O b2b;bo r b b and C b2b;bo b. Observe P Let b2b r b b P 1 (U)? that C is compact since it is a nite union of compact sets and O 2 P (UC). By proposition 2.4(ii), P (U)?. It follows by proposition 2.2, applied rst to P (U)? and then to P (UC), that ( ) O P (UC) j C. We also have O = ( ) O since O 6= by assumption. Hence, O P (UC) j C, and consequently, SÒ(f; ) = S`C (f; O) C fd. For the converse inequality, let C be a compact subset of O and let j C and j O be the restrictions of to C and to O respectively. Since f is non-negative, C fd fdj O : F By lemma 7.5, j O = O and hence fdj O = sup S`(f; O ) = sup SÒ(f; ). 2 We observe that by the above lemma the generalization to open sets that we gave for -integration extends the usual for ordinary iemann integration: indeed, if ]a; b[ is 18

19 an open interval of the real line the iemann integral of f on it can be dened in the following way: b f(x)dx lim f(x)dx = supf f(x)dx j I ]a; b[; I compactg ]a;b[!0 a+ I Therefore -integration on open sets extends usual iemann integration on open intervals. As announced, we have: Proposition 7.7 If f is a non-negative real valued function on which is continuous almost everywhere with respect to then fd = L O O fd : Proof: Since is locally compact S and second countable, there exists a chain hc i i i2in of compact sets such that O = C i2in i. By applying Lebesgue's monotone convergence theorem to the sequence of functions hf Ci i i2in, we obtain L O fd = sup L CO; C compact C fd : The conclusion then follows by lemma 7.6 and the fact that we already know that on compact spaces the Lebesgue and the generalized iemann integral coincide. 2 ecall that for any subset A of a metric space (; d) and any r 0, the r-parallel body A r of A is given by A r = fx 2 j 9y 2 A: d(x; y) rg. Proposition 7.8 If f is a non-negative real valued function on which is continuous almost everywhere with respect to then fd = L O O fd : Proof: We will start with proving that O fd L O fd. Let O 1=n be the 1=n-parallel body of O. Since L O fd = inf L n0 O 1=n fd ; it is enough to prove that, for all positive integer n, O fd L O 1=n fd. Fix n 1 and let > 0 be given. P Let M sup f[]. By proposition 5.11 there exists 1 such that, for all a2a r a a w 1, we have r a < 2M : a2a;jaj1=n By applying propositions 7.4 and 7.7 to O 1=n, there exists 2 P a2a r a a w 2, we have: a2a;ao 1=n r b sup f[b] L O 1=n fd < =2 : such that, for all 19

20 Take P a2a r a a with 1 ; 2 v. Then, O fd Su O (f; ) = P a2a;a\o6=;;jaj<1=n r a sup f[a] + P a2a;a\o6=;;jaj1=n r a sup f[a] P a2a;ao 1=n r a sup f[a] + M P a2a;jaj1=n r a P a2a;ao 1=n r a sup f[a] + =2 L O 1=n fd + : The conclusion follows since is arbitrary. As for the converse inequality, we have to prove that, for all, SO u (f; ) L O fd. For this purpose, we will use the chain of simple valuations h ii i2nconstructed in section 4 with = F i2n i. If O, there exists i 2 IN such that O i. Moreover, since, there exists j such that v j. Let k = maxfi; jg. Then O k and and v k. To x the notation, let k be the simple valuation associated to the ordered cover ho 1 ; : : :; O n i of k. Since the upper sums decrease, we have: S u (f; ) S u O (f; k) P O i\o6=; (O i [ j<i O j ) sup f[o i ] P O i\o6=; ((O i [ j<i O j ) \ O) sup f[(o i [ j<i O j ) \ O] P O i\o6=; L (O i [ j<i O j )\O fd = L O fd If O is not way-below we can use the decomposition O = S i2in O \ i, where, for all i, O \ i. By the above argument, for all i we have and therefore, for all i, that gives O\ i fd L O\ i fd fd L fd O O\ i fd sup L fd O i2in O\ i and the conclusion follows since sup i2in L O\ i fd = L O fd. 2 The following linearity properties are easily derived from the corresponding properties of the Lebesgue integral: Proposition O (f + g)d = O fd + O gd. 2. If c is any positive real number, then O cfd = c O fd. Similarly we have: 20

21 Proposition 7.10 If O 1 ; O 2 are disjoint open sets then fd = fd + O 1[O 2 O 2 O 2 fd : Proposition 7.11 If a sequence hf j i j2in of -integrable, real-valued functions dened on is uniformly convergent to the function f and O is an open subset of, then f is -integrable on O and O fd = lim j!1 O f jd. By the next proposition, in the following we can dispense with the assumption on f being non-negative. Proposition 7.12 If f is a bounded real-valued function which is continuous almost everywhere then L O fd = O f + d O f d. Proof: We have L O fd = L O f + d L O f d = O f + d O f d Computation of the -integral on an open set In order to have a computation of the integral O fd of a bounded function continuous almost everywhere with respect to, we require to have two sequences which converge to the expected value of the integral from below and from above, so that at each stage of the computation a lower bound and an upper bound for the value of the integral is obtained. First, by denition, for every simple valuation, SÒ(f; ) S u O(f; ) : Then it follows from proposition 7.2 that for any two simple valuations 1 ; 2 we have SÒ(f; 1 ) S u O(f; 2 ) and therefore sup SÒ(f; ) inf Su O(f; ) : The lower sums and the upper sums of a continuous function relative to a given a chain of valuations and a given open may not converge to the same limit, as the following example of an iterated function system (IFS) with probabilities [18, 3] will show. Example. Consider the IFS with probabilities on the space = [0; 1]: f 1 : x 7! x=2 p 1 = 1=3 f 2 : x 7! 1=2 p 2 = 1=3 f 3 : x 7! x=2 + 1=2 p 1 = 1=3 This IFS gives rise to the following chain h n i n1 of simple valuations on the upper space of, where n = 1=3 n 3 i 1;:::;i n=1 fi1 ;:::;f in [0;1] F It follows from [6, 7] that n1 n is maximal in P 1 (U) and gives the unique measure satisfying = 1=3( f f f 1 ) : 3 Observe that there is a non-zero mass on all points 1=2 n. 21

22 If we consider as f the identity function, we have, for all n, S u [0;1=2) (f; n) S`[0;1=2) (f; n) 1=6 Therefore the upper sums and the lower sums cannot converge to the same limit. The lower sums and the upper sums of f with respect to O are proved to converge to the same limit under the auxiliary assumption that the boundary of O has measure zero. Theorem 7.13 Let be a compact metric space, a normalized measure on, O an open subset of, f :! I a bounded function continuous almost everywhere with respect to. If the boundary of O has measure zero, then the lower sums and the upper sums of f on O converge to the same limit which is the Lebesgue integral of f on O. Proof: We already know that the lower sums of f on O converge from below to L O fd and that the upper sums converge from above to L O fd. If the boundary of O has fd and the conclusion follows. 2 measure zero then L O fd = L O Acknowledgements The authors wish to thank Till Plewe and Phillip Sunderhauf for their remarks on a preliminary version of this paper. eferences [1] S. Abramsky, A. Jung. Domain theory, in S. Abramsky, D.M. Gabbay and T.S. Maibaum, eds., Handbook of Logic in Computer Science, Vol. 3, Clarendon Press, Oxford, [2] T. M. Apostol. Mathematical Analysis, Addison-Wesley, [3] M. F. Barnsley. Fractals Everywhere, Academic Press, second edition, [4] G. Chaikalis. The generalized iemann integral for unbounded functions, Master Thesis, Imperial College, [5] A. Edalat. Domain Theory in Stochastic Processes, to appear in the Proceedings of Logic in Computer Science, Tenth Annual IEEE Symposium, June, 1994, San Diego, IEEE Computer Society Press, [6] A. Edalat. Dynamical systems, Measures and Fractals via Domain Theory, Information and Computation, vol 120, no. 1, pp , [7] A. Edalat. Domain Theory and Integration, Theoretical Computer Science 151, pp , [8] A. Edalat. When Scott is Weak on the Top, to appear in Mathematical Structures in Computer Science, [9] A. Edalat. Domain of computation of a random eld in statistical physics, in Proc. 2nd Imperial College, Department of Computing, Theory and Formal Methods Workshop, [10] A. Edalat. Domain Theory in Learning Processes, in Proceedings of MFPS, S. Brookes and M. Main and A. Melton and M. Mislove eds, Electronic Notes in Theoretical Computer Science, vol. 1, Elsevier, [11] A. Edalat. Power Domains and Iterated Function System, Information and Computation, pp , [12] A. Edalat and M. Escardo, Integration in eal PCF, in the proceedings of Logic in Computer Science, IEEE Computer Society,