Equiintegrability and Controlled Convergence for the Henstock-Kurzweil Integral

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1 International Mathematical Forum, Vol. 8, 2013, no. 19, HIKARI Ltd, Equiintegrability and Controlled Convergence for the Henstoc-Kurzweil Integral Esterina Mema University of Elbasan, Science Natural Faculty Mathematics Department, Elbasan, Albania esterina Copyright c 2013 Esterina Mema. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Mathematics Subject Classification: 28B05, 46G10 Keywords: Henstoc-Kurzweil integral, equi-integrability, controlled convergence theorem, generalized absolute continuity 1. Introduction and Preliminaries J.Kurzweil and J.Jarne have shown a characterization of equi-inte-grability for the Henstoc-Kurzweil integral on m-dimensional compact subintervals of R m, Main Theorem 1.1, [5]. There are also other characterizations of equiintegrability for the Henstoc-Kurzweil integral in [8] and [6]. Our characterization of equi-integrability sharpens the result in [6] (see Theorem 5 and Corollary 9, [6]). The original proofs in [6] does not consider pointwise bounded Henstoc-Kurzweil-equiintegrable sequences. We firstly get a characterization of equiintegrability for Henstoc-Kurzweil integral, Theorem 2.3. This theorem yields a controlled convergence theorem for the Henstoc-Kurzweil integral of real-valued functions, Theorem 2.4. There is a considerable wor dealing with controlled convergence theorems for Henstoc-Kurzweil integral (see e.g. [5], [3], [9], [10] etc.). The real valued version of Theorem 3.3 in [4] and Theorem 6 in [1] are other controlled convergence theorems for Henstoc-Kurzweil integral. We denote by I the family of all non-degenerate closed subintervals of [0, 1] and by λ the Lebesgue measure on [0, 1]. We will identify an interval function F : I Rwith the point function F (t) =F ([0,t]),t [0, 1]; and conversely, we will identify a point function F :[0, 1] R with the interval function

2 914 Esterina Mema F ([u, v]) = F (v) F (u), [u, v] I. An interval function F : I R is said to be additive if F (I J) =F (I) +F (J) for each nonoverlapping intervals I,J I with I J I. The intervals I,J Iare said to be nonoverlapping if int(i) int(j) =, where int(i) denotes the interior of I. A pair (I,t) of an interval I Iand a point t I is called tagged interval, t is the tag of I. A HK-partition π in [0, 1] is a finite collection of tagged intervals (I,t) with nonoverlapping intervals I. A function δ : A (0, + ) is said to be a gauge on A [0, 1]. An HK-partition π in [0, 1] is an HK-partition of [0, 1] if (I,t) π I =[0, 1], A-tagged if for all (I,t) π we have t A, δ-fine, if for every tagged interval (I,t) π we have I (t δ(s),t+δ(s)). Let us now recall a few necessary definitions. Definition 1.1. A function f :[0, 1] R is called Henstoc-Kurzweil integrable (HK-integrable) on [0, 1] if there exists w f R satisfying the following property: for every ɛ>0 there exists a gauge δ on [0, 1] such that for every δ-fine HK-partition π of [0, 1], we have f(t)λ(i) w f <ɛ. (I,t) π We write (HK) f = w S f and call w f HK-integral of f over [0, 1]. The function f is said to be HK-integrable on a set A [0, 1] if the function f.χ A :[0, 1] R is HK-integrable on [0, 1], where χ A denotes the characteristic function of A. We write (HK) f.χ [0,1] A =(HK) f for HK-integral of f on A A. If f is HK-integrable on [0, 1], then we define the function F :[0, 1] R by F (I) =(HK) f, for all I I. This function is said to be the primitive of I f on [0, 1]. By the real valued version of Theorem 3.3.5, [7], F is additive and by Theorem 7.4.1, [7], F is continuous on [0, 1]. Definition 1.2. A family M of functions f :[0, 1] R is said to be HKequiintegrable if each f Mis HK-integrable and for every ɛ > 0 there exists a gauge δ on [0, 1], such that for every δ-fine HK-partition π of [0, 1], we have f(t)λ(i) (HK) f <ɛ for all f M. (I,t) π [0,1] A function F :[0, 1] R is said to be ACG on A [0, 1] if A is the union of a sequence of subsets A i such that F is AC (A i ) for each i, i.e., for every ε>0 there is η>0such that for any finite or infinite sequence ([a,b ]) of pairwise nonoverlapping subintervals in I with at least one a,b belonging to A i satisfying (1.1) F (b ) F (a ) <ɛ,

3 Equiintegrability and controlled convergence 915 whenever (b a ) <η. Replacing (1.1) by ω(f, [a,b ]) <ɛ, where ω(f, [a,b ]) = sup{ F (b ) F (a ) :[a,b ] [a,b ]}, we obtain the definition of a function F that is ACG on A [0, 1]. A sequence of functions (F n ) is said to be UACG on A [0, 1] if A is the union of a sequence of subsets (A i ) such that (F n )isuac (A i ) for each i, that is, η in the definition of AC (A i ) with F replaced by F n is independent of n. The notion of a sequence (F n ) that is UACG (A) is similarly defined. By Lemma and 5.3.3, [10], we obtain immediately the following. Lemma 1.3. Let F :[0, 1] R be a continuous additive interval function and let A be a subset of [0, 1]. Then F is AC (A) if and only if F is AC (A). The sequence (F n ) is said to satisfy the uniformly strong Lusin condition, or briefly USL, if for every ɛ>0 and every subset Z S with λ(z) = 0 there exists a gauge δ on Z such that for each δ-fine Z-tagged HK-partition π in [0, 1], we have (I,t) π F n(i) <ɛfor all n N. 2. The main results The main theorems are Theorem 2.3 and Theorem 2.4. The following lemmas mae it possible to present clearly Theorem 2.3. The notion of HKequiintegrability does not allow one to ignore sets of measure zero (see [2]). Nevertheless, Lemma 2.1 allows us to ignore sets of measure zero in the notion of HK-equiintegrability. The only if part of the lemma is proven in [2], Exercise The if part is straightforward. Lemma 2.1. Let (f n ) be a pointwise bounded sequence of functions f n :[0, 1] R and let E be subset of [0, 1] such that λ(s \ E) =0. Then, the sequence (f n ) is HK-equiintegrable if and only if the sequence (f n.χ E ) is HK-equiintegrable. The next lemma shows that the condition (f n )ishk-equiintegrable implies that pointwise bounded and USL notions are equivalent. Lemma 2.2. Let (f n ) be a sequence of functions f n :[0, 1] R and let f : [0, 1] R be a function. If we have f n (s) f(s) almost everywhere in [0, 1], (f n ) is HK-equiintegrable, then the following statements are equivalent (i) (f n ) is pointwise bounded, (ii) (F n ) is USL. Proof. By hypothesis, there exists Z [0, 1] with λ(z) = 0 such that (f n ) converges pointwise in E =[0, 1] \ Z to f.

4 916 Esterina Mema Assume that (f n ) is pointwise bounded and denote f (E) = f.χ E, f n (E) = f n.χ E, n N. Then, the sequence (f n (E) ) converges pointwise in [0, 1] to f (E) and by Lemma 2.1 the sequence (f n (E) )ishk-equiintegrable. By Theorem 3.3.7, [7], we get also that each F n is the primitive of HK-integrable function f n (E). Therefore, we obtain by Theorem 8, [6] that (F n )isusl. Conversely, we assume that (F n )isusl. Let z be an arbitrary element of Z. Since (f n )ishk-equiintegrable, by Lemma 3.5.6, [7], for the given ɛ>0 there exists a gauge δ 1 on Z such that for every δ 1 -fine Z-tagged HK-partition π in [0, 1], we have (f n (s)λ(i) F n (I)) ɛ for all n N. (I,s) π Since (F n )isusl, there exists a gauge δ 2 on Z such that for every δ 2 -fine Z-tagged HK-partition π in [0, 1], we have (I,s) π F n(i) <ɛ,for all n N. Now, if we denote δ = min{δ 1,δ 2 } and consider any δ-fine {z}-tagged HKpartition π = {(I,z)} in [0, 1], then we get f n (z)λ(i) f n (z)λ(i) F n (I) + F n (I) 2.ɛ, for all n N. Thus, we have f n (z) 2ɛ = M>0, for all n N. Since z λ(i) was arbitrary, the last result yields that (f n ) is pointwise bounded. Theorem 2.3. Let (f n ) be a sequence of HK-integrable functions f n :[0, 1] R and let f :[0, 1] R be a function. If we have f n (s) f(s) almost everywhere in [0, 1], (F n ) is USL, where F n s are the primitives of f n s, then the following statements are equivalent (A) every subsequence of (f n ) has a subsequence HK-equiintegrable, (B) every subsequence of (F n ) has a subsequence UACG (W ), where W [0, 1] such that λ([0, 1] \ W )=0. Proof. (A) (B) Let us consider any subsequence (G n ) (F n ). We denote by (g n ) the subsequence of (f n ) which is the sequence corresponding to (G n ). For the subsequence (g n ) (f n ) there exists a subsequence (h n ) (g n ) which is HK-equiintegrable. Let (H n ) be the subsequence of (G n ) which is the sequence corresponding to (h n ). Therefore by Corollary 9, [6], the subsequence (H n ) (G n )isuacg (W ), where W [0, 1] such that λ([0, 1] \ W ) = 0. Thus, (H n ) is the required subsequence of (G n ). (B) (A) Let (g n ) be any subsequence of (f n ). We denote by (G n ) the subsequence of (F n ) which is the sequence corresponding to (g n ). For the subsequence (G n ) (F n ) there exists a subsequence (H n ) (G n ) which is UACG (W ), where W [0, 1] such that λ([0, 1] \ W ) = 0. Let (h n ) be the subsequence of (g n ) which is the sequence corresponding to (H n ). Hence, by Theorem 5 in [6], for the sequence (h n ) there exists a sequence (h n ( ) ) such that

5 Equiintegrability and controlled convergence 917 h ( ) n (s) =h n (s) almost everywhere in [0, 1] for all n N, there is a HK-equiintegrable subsequence (h n ( ) ) (h n ( ) ). Then, there exists Z [0, 1] with λ(z) = 0 such that h n ( ) i (s) =h ni (s) for every s E =[0, 1] \ Z and i N, and consequently the sequence (h n ( ) (s)) converges to f(s) for each s E. Hence, by Theorem and Theorem in [7], for each I I, we have (HK) h ( ) n =(HK) h n = H n (I), I I and since (H n )isusl and (h n ( ) ) converges pointwise in E =[0, 1] \ Z to f, we get by Lemma 2.2 that (h n ( ) ) is pointwise bounded. Consequently, we obtain by Lemma 2.1 that (h n )ishk-equiintegrable. Thus, (h n ) is required subsequence of (g n ) and the proof is finished. Theorem 2.4. If a sequence (f n ) of HK-integrable functions f n : S R and a function f : S R satisfy the following conditions (i) f n (s) f(s) almost everywhere in [0, 1], (ii) (F n ) is USL, where F n s are the primitive of f n s, (iii) (F n ) satisfies the condition (B) from Theorem 2.3, then the function f is HK-integrable on [0, 1] and for every I Iwe have lim n F n (I) =F (I), where F is the primitive of f. Proof. We firstly show the theorem for the case when I =[0, 1]. Let (g n )be any subsequence of (f n ). Then, we obtain by Theorem 2.3 that there exists a subsequence (h n ) (g n ) such that (h n )ishk-equiintegrable. By hypothesis there exists Z [0, 1] with λ(z) = 0 such that (h n ) converges pointwise in E = S \ Z to f. Denote f (E) = f.χ E and h n (E) = h n.χ E,n N. Then, the sequence (h n (E) ) converges pointwise in [0, 1] to f (E). By Lemma 2.2 and Lemma 2.1 we have also that (h n (E) ) ishk-equiintegrable. Therefore, by Theorem in [7], the function f (E) is HK-integrable on [0, 1] and (2.1) lim (HK) h n n [0,1] (E) =(HK) f (E). [0,1] Hence, we obtain by Theorem in [7] that the function f is HK-integrable on [0, 1] and lim n (HK) h [0,1] n =(HK) f. Since every subsequence [0,1] (g n ) (f n ) has a subsequence (h n ) (g n ) satisfying the above property, the sequence itself has the property. Thus, we have lim n F n ([0, 1]) = F ([0, 1]), where F is the primitive of f on [0, 1]. Now, we show the theorem for an arbitrary I I. Denote f (I) = f.χ I and f n (I) = f n.χ I,n N. Then, by Theorem in[7],f (I) and each f n (I) is HK-integrable on [0, 1]. Note that { F n (I) (J) = Fn (I J) I J 0 I J =

6 918 Esterina Mema for every J I, where F n (I) s are the primitives of f n (I) s. Evidently (f n (I) ) satisfies (i). Let us show that (F n (I) ) satisfies (ii) and (iii). Firstly, we show that (F n (I) )isusl. By (I) the set of endpoints of I is denoted. Let us consider any ɛ>0 and an arbitrary Z [0, 1] with λ(z) =0. We can define a gauge δ 0 on Z such that for every δ 0 -fine Z-tagged HKpartition π in [0, 1], we have (J, z) π and z/ I J I or J S \ I Since (F n )isusl, there exists a gauge δ 1 on Z such that for each δ 1 -fine Z-tagged HK-partition π in [0, 1], we have (J,z) π F n(j) < ɛ, for all n N. 3 Hence, we obtain F n (I) (J) = F n (I) (J) + F n (I) (J) = (J,z) π z (I):(J,z) π z (I):(J,z) π F n (I) (J) + z int(i):(j,z) π z/ (I):(J,z) π F n (J) < 2. ɛ 3 + ɛ 3 = ɛ for each δ-fine Z-tagged HK-partition π = {(J, z)} in [0, 1], where δ = min{δ 0,δ 1 }. This means that (F n (I) )isusl. Secondly, since each F n (I) is continuous on [0, 1] and for every J Iwe have,j) ω(f n,j), we obtain by Lemma 1.3 that (F (I) ) satisfies (iii). ω(f (I) n Thus, by applying the same arguments as in the case when I =[0, 1] we infer lim n F I n(i) =F (I). Finally, since I was arbitrary we get lim n F n (I) = F (I), for every I I. Acnowledgement. Author thans Professor Sool Kaliaj for the preparation of this paper. References [1] L. Di Piazza and K. Musial, Characterizations of Kurzweil-Henstoc-Pettis integrable functions, Stud. Math. 176, No.2, (2006). [2] R. Gordon, Another loo at a convergence theorem for the Henstoc integral, Real Anal. Exchange 15, No.2, (1990). [3] R. Henstoc, Lectures on the Theory of Integration, World Scientific, Singapore, [4] S. B. Kaliaj, A. D. Tato, F. D. Gumeni, Controlled convergence theorems for Henstoc- Kurzweil-Pettis integral on m-dimensional compact intervals, Czechoslova Math. J. 62 (2012), [5] J. Kurzweil and J. Jarni, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exchange, Vol.17 ( ). [6] W. Pujie, Equi-integrability and controlled convergence for the Henstoc integral Real Anal. Exchange, Vol.19 ( ). [7] Š. Schwabi and Y. Guoju, Topics in Banach Space Integration, Series in Real Analysis [8] 10, World Scientific, Singapore (2005). Š. Schwabi, Henstoc s Condition for Convergence Theorems and Equi-integrability, Real Anal. Exchange, Vol.18, [9] L. P. Yee, Generalized convergence theorems for Denjoy-Perron integrals, In: New Integrals, Ed. P. S. Bullen et all., Lecture Notes in Math. 1419, Springer Berlin, n

7 Equiintegrability and controlled convergence 919 [10] L. P. Yee and R. Vyborny, Integral: An Easy Approach after Kurzweil and Henstoc, Cambridge University Press, First published (2000). Received: March 17, 2013