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1 International Publications (USA PanAmerican Mathematical Journal Volume 16(2006, Number 4, he Kurzweil-Henstock Integral for Riesz Space-Valued Maps Defined in Abstract opological Spaces and Convergence heorems Antonio Boccuto Università degli Studi di Perugia Dipartimento di Matematica e Informatica via Vanvitelli, 1 I Perugia, Italy boccuto@yahoo.it Beloslav Riečan Univerzita Mateja Bela ajovského 40 SK Banská Bystrica, Slovakia riecan@fpv.umb.sk Communicated by Carlo Bardaro (Received May 2006; Accepted June 2006 Abstract We prove some convergence theorems for the Kurzweil-Henstock integral in Riesz spaces, for functions defined in a compact topological space with respect to Riesz spacevalued measures, which can assume even the value AMS Subject Classification: 28B15, 28B05, 28B10, 46G10. Key words and phrases: Kurzweil-Henstock integral, Riesz spaces, product triple, convergence theorems. 1 Introduction In this paper we deal with convergence theorems (monotone convergence theorem and Lebesgue dominated convergence theorem for the Kurzweil-Henstock integral in an abstract setting, for functions defined in a compact topological space, satisfying suitable properties, and with values in a Dedekind complete Riesz space, with respect to a positive Riesz space-valued measure µ, which can assume even the value. A particular case is =[A, B] an interval (possibly unbounded of the extended real line, or the whole of R,

2 64 A. Boccuto and B. Riečan and µ = the Lebesgue measure (this case was investigated in [2]. We continue the investigation started in [4], in which µ is R-valued, and in which some other kinds of convergence theorems were demonstrated. Our results given here extend the ones in [14], Chapter 5, which were proved in the case in which µ( is finite, and the ones of [9], which were proved in the case =[A, B], where [A, B] is as above, and all the involved Riesz spaces coincide with the (eventually extended real line. A similar Kurzweil-Henstock type integral was investigated in [5] for Banach space-valued maps. his approach can be used in several contexts: for example, in the study of states and observables (see [14], which can be viewed as functions or probability measures with values in special Riesz spaces, and in the theory of stochastic processes, which can be considered as L 0 -valued maps, where the involved convergence is the almost everywhere convergence, which does not have a topological nature. 2 Preliminaries Assumption 2.1. We assume that (, is a Hausdorff compact topological space, and that there are given a family B of Borel subsets of such that Band closed under intersections, and a monotone positive additive mapping µ : B R (here, additive means that µ(e F µ(e F =µ(eµ(f whenever E,F,E F B. Definition 2.2. A gauge on is a mapping γ : assigning to each t its neighborhood γ(t. Example 2.3. Let =[a, b] R, δ : R be a map. Let be the usual topology on the real line. Put γ(t =(t δ(t,t δ(t. hen γ is a gauge in the sense of Definition 2.2. Definition 2.4. A partition of is a finite collection = {(E i,t i :i =1,...,k} of couples such that (i k E i = ; (ii t i E i,e i B; (iii µ(e i E j =0(i j. A collection satisfying axioms (ii and (iii, but not necessarily (i, is called decomposition of. he partition or decomposition is γ-fine ( γ, if E i γ(t i (i =1, 2,...,k. Definition 2.5. We say that B is separating, if there exists a sequence ( n n of partitions such that n1 is a refinement of n (n N and, for any x, y,x y, there exist n and E n such that, if x E for some E n, then y \ E. Let R be a Dedekind complete Riesz space. definitions. We now give the following preliminary

3 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 65 Definition 2.6. A bounded double sequence (a i,j i,j in R is called a (D-sequence if a i,j a i,j1 i, j N and a i,j =0 i N. j=1 Definition 2.7. A sequence (p n n is called an (O-sequence if p n p n1 n N and p n = 0. In this case, we write also p n 0. n=1 Definition 2.8. Given a sequence (r n n in R, we say that (r n n (D-converges to an element r R if there exists a (D-sequence (a i,j i,j in R, satisfying the following condition: ϕ N N, there exists an integer n 0 such that r n r a i,ϕ(i, for all n n 0. In this case, we write (D lim n r n = r. Definition 2.9. Given a sequence (r n n in R, we say that (r n n (O-converges to an element r R if there exists an (O-sequence (p n n in R, such that r n r p n n N. In this case, we write (O lim n r n = r. We now recall the very famous Fremlin Lemma (see also [6, 14]: Lemma Let (a (k i,j i,j, k N, be any countable family of (D-sequences. hen for each fixed element u R, u 0, there exists a (D-sequence (a i,j i,j such that, for every ϕ N N, one has u ( s k=1 a (k i,ϕ(ik a i,ϕ(i s N. Definition A Riesz space R is said to be weakly σ-distributive if for every bounded double sequence (b i,j i,j with b i,j b i,j1 i, j one has: b i,j = ( b i,ϕ(i (1 j=1 ϕ N N (see [19]. he following characterization holds (see also [7], Proposition 2.1, pp Proposition A Dedekind complete Riesz space R is weakly σ-distributive if and only if for every (D-sequence (a i,j i,j in R one has: ( a i,ϕ(i =0. (2 ϕ N N

4 66 A. Boccuto and B. Riečan (his implies that in weakly σ-distributive spaces (D-limits are unique From now on, we shall always suppose that R is a Dedekind complete weakly σ-distributive Riesz space. Definition Let S be the σ-algebra of all Borel subsets of. We say that µ : S R is regular, if to any E Sthere exists a (D-sequence (a i,j i,j such that for every ϕ : N N there exist C compact and U open such that C E U and sup B S,B U\C µ(b a i,ϕ(i. Remark If (,d is a compact metric space, B is a semiring of subsets of such that to any x and every neighborhood U of x there exists E Bsuch that x int E E U (where, given any subset E of any topological space, we denote by int E its interior in the topological sense and B is separating, then to every gauge γ : there exists a γ-fine partition and in correspondence with any set B Bthere exists a γ B -fine partition (see also [11], Lemma 1.2., p. 154 and Proposition 1.7., p From now on, we shall always suppose that B satisfies conditions in We now introduce some structural assumptions, which will be needed later. Assumptions Let R 1, R 2, R be three Dedekind complete Riesz spaces. We say that (R 1,R 2,Risaproduct triple if there exists a map : R 1 R 2 R, which we will call product, such that (r 1 s 1 r 2 = r 1 r 2 s 1 r 2, r 1 (r 2 s 2 =r 1 r 2 r 1 s 2, [r 1 s 1,r 2 0] [r 1 r 2 s 1 r 2 ], [r 1 0, r 2 s 2 ] [r 1 r 2 r 1 s 2 ] for all r j,s j R j, j =1, 2; if r j R j, j =1, 2 and α R, then (αr 1 r 2 = r 1 (αr 2 =α(r 1 r 2 ; if (a λ λ Λ is any net in R 2 and b R 1, then [a λ 0, b 0] [b a λ 0]; if (a λ λ Λ is any net in R 1 and b R 2, then [a λ 0, b 0] [a λ b 0]. We will write often ab instead of a b. A Dedekind complete Riesz space R is called an algebra if (R, R, R is a product triple. We always assume that (R 1,R 2,R is a product triple and that R is weakly σ-distributive. Furthermore, we add to R 2 two extra elements, and, extending ordering and operations in a natural way, and denote R 2 = R 2 {, }.

5 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 67 3 he Kurzweil-Henstock integral We now give our definition of Kurzweil-Henstock integrability. We suppose that µ : S R 2 is an additive positive regular measure on the Borel σ-algebra S of.if={(e i,t i :i = 1,...,q} is a partition or a decomposition of a set A B, and f : R 1, then we define the Riemann sum as follows: q f = f(t i µ(e i if the sum exists in R, with the conventions that 0 ( = 0 and it is not considered any contribution in which µ(e i =. he points t i, i =1,...,q, are called tags. Definition 3.1. A function f : R 1 is (KH-integrable (or, in short, integrable if there exist I R and a (D-sequence (a i,j i,j such that ϕ N N there exists a gauge γ such that f I a i,ϕ(i (3 whenever = {(E i,t i,,...,q} is a γ-fine partition of. Evidently the number I is determined uniquely. It will be denoted by (KH f dµ, fdµor f dµ. Proposition 3.2. he integral is a linear positive functional. Proof: he linearity follows by the identity (αf βg =α he positivity follows by the implication f 0 f β f 0 g. and weak σ-distributivity of R. Definition 3.3. A function f : R 1 is (KH-integrable (or, in short, integrable on a set E Bif there exist I E R and a (D-sequence (a (E i,j i,j such that ϕ N N there exists a gauge γ such that f I E a (E i,ϕ(i (4 whenever = {(E i,t i,,...,q} is a γ-fine partition of E.

6 68 A. Boccuto and B. Riečan he element I E is denoted by fdµ. When we say simply integrable, we will intend E integrable on. We now state the Bolzano-Cauchy condition. heorem 3.4. A map f : R 1 is (KH-integrable on E Bif and only if there exists a (D-sequence (b (E i,j i,j such that, ϕ N N, a gauge γ such that for all γ-fine partitions 1, 2 of E we have f f b (E i,ϕ(i. 1 he proof is similar to the one of [14], Proposition 5.2.9, pp Proposition 3.5. If E,F,G B, E = F G, µ(f G =0and f is integrable on E, then f is integrable on F and on G too, and fdµ= fdµ f dµ. E F G he proof is similar to the one of [14], Proposition , pp By induction, it is possible to prove the following: Proposition 3.6. If E B, E i B, i =1,...,n, and E = n E i, µ(e i E j =0 whenever i j and f is integrable on E, then f is integrable on E i for every i, and n fdµ= f dµ. E E i Of course, if we want to get an integral corresponding with our intuition, the value of the integral of a simple function of the type n α iχ Gi should be n α iµ(g i, if µ(g i R 2 for every i =1,...,n. In order to do this, it is enough to prove the following theorem (see also [14], Proposition , pp : heorem 3.7. Let S be the class of all Borel sets of, µ : S R 2 be positive, additive and regular, E S, with µ(e R 2. Let r R 1, and g = χ E r (defined by the relation g(t =r, ift E, and g(t =0,ift E. hen g is integrable, and gdµ= rµ(e. Proof: First of all, assume r 0. By regularity of µ on S there exists a (D-sequence (a i,j i,j such that for every ϕ N N there exist an open set U and a compact set C, C E U, such that µ(u \ C a i,ϕ(i.

7 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 69 Since C is compact and U is open, there exists a gauge η such that η(t U t C, η(t U \ C t U \ C, η(t C = t U. ake any partition η, ={(E i,t i : i =1,...,n}. hen rµ(c rµ(e rµ(u, rµ(u \ C =rµ(u rµ(c, and rµ(e r a i,ϕ(i rµ(u r a i,ϕ(i rµ(c ( rµ E i rµ(e i and hence = t i C t i C n n χ C (t i rµ(e i χ E (t i rµ(e i = g rµ(e i rµ(u rµ(er a i,ϕ(i, t i U g rµ(e r a i,ϕ(i for any partition η. From the properties of the product it follows that the double sequence (r a i,j i,j is a (D-sequence. hus, we get the assertion, at least when r 0. In the general case (r R 1 we get: χ E rdµ= χ E (r r dµ = χ E r dµ χ E r dµ 4 Convergence theorems = r µ(e r µ(e =rµ(e. We begin with proving a theorem, which will be used in the sequel, in order to demonstrate our versions of convergence theorems. heorem 4.1. Let (f n : R 1 n be a sequence of integrable functions. Suppose that: there is a (D-sequence (b i,j i,j such that to every ϕ : N N there exist a gauge ζ and n 0 N such that f n dµ f n b i,ϕ(i for every partition ζ and n n 0 ;

8 70 A. Boccuto and B. Riečan there exist a function f : R 1,a(KH-integrable map h : R (with respect to µ and a (D-sequence (a i,j i,j such that, ϕ N N, t, p(t N: n p(t, f n (t f(t h (t. (5 hen f is integrable and (D lim n f n dµ = a i,ϕ(i f dµ. Proof: We shall use the Bolzano-Cauchy condition. Let (b i,j i,j, ζ and n 0 be as in By we get the existence of an element 0 w R 2 such that ϕ N N there exists a gauge η ζ (without loss of generality such that, for every η-fine partition of,={(e i,t i :i =1,...,q}, we have: f f n [f(t i f n (t i ]µ(e i h (t i µ(e i (6 a i,ϕ(i w a i,ϕ(i n max{p(t i :i =1,...,q}. Put a i,j = a i,j w, i, j N. Without loss of generality, we can suppose that p(t i n 0 i =1,...,n. Choose now a (D-sequence (c i,j i,j such that 2 a i,ϕ(i b i,ϕ(i c i,ϕ(i. hen for all partitions 1, 2 η, we have (using sufficiently large n, depending on the involved partitions 1 and 2 : f f f f n f n f n dµ f n dµ f n 1 2 f n f c i,ϕ(i. 2 he integrability of f follows from this and the Bolzano-Cauchy condition. 2

9 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 71 By integrability of f we obtain the existence of a (D-sequence (a i,j i,j such that, for every ϕ N N, there exists a gauge η 1, depending on ϕ, such that fdµ f a i,ϕ(i1 for every partition η 1. By there is a (D-sequence (b i,j i,j such that f k f k dµ b i,ϕ(i2 for every k greater than a suitable integer k 0 (depending on the involved ϕ and for each partition η 2. By 4.1.2, proceeding as in (6, we get the existence of a (D-sequence (c i,j i,j such that f f k c i,ϕ(i3 for every k k, where k is a positive integer depending on the involved partition. Without loss of generality, we can assume k k 0. Choose a (D-sequence (d i,j i,j such that a i,ϕ(i1 b i,ϕ(i2 c i,ϕ(i3 d i,ϕ(i. hen ( being a partition chosen arbitrarily, η 1 η 2 fdµ f k dµ fdµ f f f k f k f k dµ for every k k. We have proved that (D lim k f k dµ = fdµ with respect to the (D-sequence (d i,j i,j. his concludes the proof. We now prove a version of the Henstock lemma. d i,ϕ(i heorem 4.2. Let g : R 1 be an integrable function. Let (a i,j i,j be a (D-sequence such that to every ϕ : N N there exists a gauge η such that gdµ g a i,ϕ(i

10 72 A. Boccuto and B. Riečan for every partition η. If ={(E i,t i :i =1,...,n} η, then for every L, L {1,...,n}, we have: gdµ g(t i µ(e i i L E i a i,ϕ(i. i L Proof: First of all, we note that integrability of g on all E i s follows from Proposition 3.6. So there exists a (D-sequence (b i,j i,j such that to every ψ : N N there is a gauge η such that gdµ g i L E i i L b i,ψ(i whenever i η E i, i L. Put η i =(η Ēi η,take i η i, i L, and set = {(E i,t i : i L} ( i L i. hen η, hence gdµ g a i,ϕ(i, and Now Since = i i gdµ g i L E i i L b k,ψ(k. k=1 gdµ g(t i µ(e i i L E i i L gdµ gdµ g g i L E i i L i gdµ g g gdµ i L i L E i i a i,ϕ(i b k,ψ(k. k=1 gdµ g(t i µ(e i i L E i a i,ϕ(i i L k=1 b k,ψ(k for every ψ : N N, by weak σ-distributivity of R we obtain: gdµ g(t i µ(e i i L E i a i,ϕ(i 0. i L Finally, we are ready to prove the monotone convergence theorem.

11 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 73 heorem 4.3. Let (f( n : R 1 n be a sequence of integrable functions, f n f n1 (n N, and let the sequence f n dµ be bounded. Suppose that: n there exist a function f : R 1,a(KH-integrable map h : R (with respect to µ and a (D-sequence (a i,j i,j such that, ϕ N N, t, p(t N: n p(t, Furthermore, assume that f(t f n (t h (t a i,ϕ(i. ( there exist a R, a 0, and a gauge γ, such that, for every γ-fine partition of, we have: f n f n dµ a n N. hen f is integrable on, and fdµ=(d lim n f n dµ. Remark 4.4. We note that regularity does imply σ-additivity of µ, at least for R 2 -valued positive finitely additive measures. Moreover we observe that, if needed, a distinguished point x 0 could be dropped from, provided it is possible to define f(x 0 =f n (x 0 =0,n N, without affecting the other hypotheses. Furthermore, we observe that, when R 1 = R 2 = R = R and =[A, ] or =[,B] is a halfline of the extended real line, condition is equivalent to pointwise convergence of the sequence (f n n to f: indeed, it is sufficient to take the function h defined by setting h (t = 1, t [A, ort (,B]. 1t2 ( Proof of heorem 4.3: Since the sequence f n dµ is bounded and increasing, it admits the (D-limit in R. hus, there exists a (D-sequence (c i,j i,j in R such that, for every ϕ N N, there exists k 0 N such that, k, l k 0, f k dµ f l dµ c i,ϕ(i. (8 Furthermore, from we get the existence of an element 0 w R 2 such that ϕ N N there exists a gauge γ such that, for every γ -fine partition of,={(e i,t i :i = n

12 74 A. Boccuto and B. Riečan 1,...,q}, we have: [f(t i f p(ti(t i ] µ(e i h (t i a i,ϕ(i µ(e i a i,ϕ(i w. (9 Note that in (9 the natural numbers p(t i can be chosen greater than k 0. Since f k is integrable k N, then for each k N there exists a (D-sequence (a (k i,j i,j such that, for every ϕ N N, there exists a gauge γ k such that for every partition γ k we have f k f k dµ a (k i,ϕ(ik1. (10 For each i, j N, put b (1 i,j =2a i,j w, and b (m i,j = a (m 1 i,j (m =2, 3,... Moreover, let a be as in By virtue of the Fremlin lemma there exists a (D-sequence (b i,j i,j such that, ϕ N N and s N, ( s a b i,ϕ(i. (11 m=1 a (m i,ϕ(im Let ϕ N N and k 0 = k 0 (ϕ be as in (8. Put γ 0 (t =γ (t γ(t γ 1 (t γ 2 (t... γ p(t (t, where the involved gauges are the ones associated with ϕ, as above. Choose a partition γ 0,={(E i,t i :i =1,...,q}. Fix arbitrarily k>k 0, where k 0 is as in (8. We have: f k f k dµ f k (t i µ(e i f k dµ p(t i k p(t E i k i f k (t i µ(e i f k dµ E i. (12 p(t i<k p(t i<k Construct similarly as in the proof of the Henstock lemma, i.e. ={(E i,t i :kp(t i } i, p(t i<k where i is a sufficiently fine partition of E i, in such a way that γ k. hen f k f k dµ a (k i,ϕ(ik1.

13 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 75 Hence, by the Henstock lemma (L = {i : p(t i k} and L = {i : p(t i =k}, we obtain f k (t i µ(e i f k dµ E i a (k i,ϕ(ik1 (13 p(t i k p(t i k and p(t i=k f k (t i µ(e i p(t i=k f k dµ E i a (k i,ϕ(ik1 (14 respectively. We now estimate the second part of the right side of (12. We have: f k (t i µ(e i f k dµ p(t i<k p(t E i<k i f k (t i µ(e i f p(ti(t i µ(e i m=k 0 p(t i=m m=k 0 p(t i=m f p(ti(t i µ(e i f p(ti dµ m=k 0 p(t i=m m=k 0 p(t E i=m i (f k f m dµ E i = m=k 0 p(t i=m m=k 0 p(t i=m m=k 0 p(t i=m m=k 0 p(t i=m k m=1 (f k (t i f p(ti(t i µ(e i (15 f m (t i µ(e i b (1 i,ϕ(i1 b (1 i,ϕ(i1 E i (f k f k0 dµ m=k 0 k m=2 b (m i,ϕ(im p(t i=m a (m i,ϕ(im1 b (m i,ϕ(im f m dµ E i (f k f k0 dµ. (f k f k0 dµ (f k f k0 dµ hus, from 4.3.2, (8, (11 and (15 we get the existence of a (D-sequence (d i,j i,j such that, for every ϕ N N, there exist a gauge γ 0 and k 0 N such that, for each γ 0 -fine

14 76 A. Boccuto and B. Riečan partition and k>k 0, we have: f k f k dµ d i,ϕ(i. (16 he assertion follows from Lemma 4.1. We now state and prove a version of the Lebesgue dominated convergence theorem. heorem 4.5. Let (f n : R 1 n be a sequence of integrable functions, and suppose that κ : R 1 is an integrable map, such that f n (x κ(x for all x and n N. Suppose that: there exist a function f : R 1,a(KH-integrable map h : R (with respect to µ and a (D-sequence (a i,j i,j such that, ϕ N N, t, p(t N: n p(t, f n (t f(t h (t a i,ϕ(i. (17 hen f is integrable and Proof: For all s N and k s, put moreover, for each s N, set g s,k = fdµ=(d lim n n,m s,min(n,mk g s = n,m s f n dµ. f n f m. f n f m ; We shall prove that, for each fixed s N, the sequence (g s,k k s satisfies the hypothesis of heorem 4.3. First of all, it is easy to check that the sequence ( g s,k dµ is well-defined and bounded in R (Indeed, it is possible to check that the g s,k s are integrable, taking into account that κ is integrable and proceeding analogously as in [1], heorem 4.33 and [8], Lemma 4.2. Let now h and (a i,j i,j be as in We know that, ϕ N N, t, s N, p N, with p s, such that, k p, ( f n (t f m (t 2 h (t a i,ϕ(i. (18 n,m k k

15 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 77 Fix arbitrarily s N. We have, for every t : f n (t f m (t = and hence n,m s n,m s,min(n,mk n,m s,min(n,mk 0 g s (t g s,k (t f n (t f m (t f n (t f m (t n,m k n,m s,min(n,m k n,m k f n (t f m (t, f n (t f m (t f n (t f m (t k s, t. hus, we get that the (D-sequence (a i,j i,j is such that, ϕ N N, t, s N, p N, with p s, such that, k p, g s (t g s,k (t 2 h (t a i,ϕ(i. So, is satisfied. We now turn to As κ is integrable, there exist a gauge γ and a positive element a R such that, for every γ-fine partition = {(E i,t i :i =1,...,q}, s N, k s, we get: q f n (t i f m (t i µ(e i that is 2 n,m s,min(n,mk q κ(t i µ(e i 2 a, (19 q g s,k (t i µ(e i 2 a. From this it follows that is satisfied. hus we get that, for every s N, g s is integrable and g s dµ = g s,k dµ. k s We now prove that the sequence ( g s s satisfies the( hypotheses of heorem 4.3. First of all, it is easy to check that the sequence g s dµ is bounded. Furthermore, we know that ϕ N N, t, p N such that, s p, ( f n (t f m (t h (t a i,ϕ(i, n,m s s

16 78 A. Boccuto and B. Riečan that is g s (t = g s (t h (t a i,ϕ(i. So, is satisfied. Concerning 4.3.2, it is enough to check that the argument in (19 works even if we replace f n (t i f m (t i with f n (t i f m (t i. hus, we get that (D lim s n,m s,min(n,mk g s dµ = s N n,m s g s dµ =0. (20 Proceeding analogously as in the proof of heorem 4.3, it is possible to prove the existence of (D-sequences (e (m i,j i,j, m N, such that, ϕ N N, there exist a gauge γ and k N such that, for each γ -fine partition = {(E i,t i,,...,q}, k>k, we have: f k f k dµ ( k e (m i,ϕ(im (f k f m dµ (21 m=1 m=k p(t E i=m i ( k g k dµ. m=1 e (m i,ϕ(im From (21 we get the existence of a (D-sequence (d i,j i,j such that, ϕ N N, there exist a gauge γ and k N such that, for each γ -fine partition, k>k, we have: f k f k dµ d i,ϕ(i. (22 he assertion follows from (22 and heorem 4.1. References [1] A. Boccuto, Differential and Integral Calculus in Riesz spaces, atra Mountains Math. Publ. 14 (1998, [2] A. Boccuto and B. Riečan, On the Henstock-Kurzweil integral for Riesz-space-valued functions defined on unbounded intervals, Czech. Math. J. 54 (129 (2004, [3] A. Boccuto and B. Riečan, A note on the improper Kurzweil-Henstock integral, Acta Univ. Mathaei Belii, Ser. Math. 9 (2001, 7-12.

17 he Kurzweil-Henstock Integral for Riesz Space-Valued Maps 79 [4] A. Boccuto and B. Riečan, A note on improper Kurzweil-Henstock integral in Riesz spaces, Acta Math. (Nitra 5 (2002, [5] A. Boccuto and A. R. Sambucini, he Henstock-Kurzweil integral for functions defined on unbounded intervals and with values in Banach spaces, Proceedings of the II Conference of Mathematics in Nitra, Acta Math. (Nitra 7 (2004, [6] D. H. Fremlin, A direct proof of the Matthes-Wright integral extension theorem, J. London Math. Soc. 11 (1975, [7] J. Jakubík, Weak σ-distributivity of lattice ordered groups, Math. Bohemica 126 (2001, [8] P. Y. Lee, Lanzhou Lectures on Henstock integration, World Scientific Publishing Co., [9] L. P. Lee and R. Výborný, he integral: An easy approach after Kurzweil and Henstock, Cambridge Univ. Press, [10] W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, I, North-Holland Publishing Co., [11] B. Riečan, On the Kurzweil Integral in Compact opological Spaces, Radovi Mat. 2 (1986, [12] B. Riečan, On the Kurzweil integral for functions with values in ordered spaces, I, Acta Math. Univ. Comenian (1989, [13] B. Riečan, On operators valued measures in lattice ordered groups, Atti Sem. Mat. Fis. Univ. Modena 40 (1992, [14] B. Riečan and. Neubrunn, Integral, Measure and Ordering, Kluwer Acad. Publ./Ister Science, [15] B. Riečan and M. Vrábelová, On the Kurzweil integral for functions with values in ordered spaces, II, Math. Slovaca 43 (1993, [16] B. Riečan and M. Vrábelová, On integration with respect to operator valued measures in Riesz spaces, atra Mountains Math. Publ. 2 (1993, [17] B. Riečan and M. Vrábelová, On the Kurzweil integral for functions with values in ordered spaces, III, atra Mountain Math. Publ. 8 (1996, [18] B. Riečan and M. Vrábelová, he Kurzweil construction of an integral in ordered spaces, Czechoslovak Math. J. 48 (123 (1998, [19] J. D. M. Wright, he measure extension problem for vector lattices, Ann. Inst. Fourier, Grenoble 21 (1971,