A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions

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1 Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, HIKARI Ltd, A Convergence Theorem for the Improper Riemnn Integrl of Bnch Spce-vlued Functions Jun Alberto Escmill, Mrí Gudlupe Rggi, Luis Ángel Gutiérrez nd Fernndo Hernández Fcultd de Ciencis Físico Mtemátics Benemérit Universidd Autónom de Puebl Puebl, México Copyright c 2014 Jun Alberto Escmill et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct In this pper, we estblish chrcteriztion, by mens of the uniform convergence, of convergence which is bsed in the Moore s ides [4]; then, ccording to this chrcteriztion, we give convergence theorem for the improper Riemnn integrl of functions tking vlues in Bnch spce. Mthemtics Subject Clssifiction: 28B05, 46G12 Keywords: Improper Riemnn integrl; Riemnn integrl; uniform convergence; m-topology 1 Introduction Throughout this pper, the symbol A will denote to subset no empty of rel numbers set R nd, the symbols f nd (fn) will denote, respectively, to vector function nd to sequence of vector functions defined on A, unless otherwise stted; i.e., f, f n : A X, where X is Bnch spce. A clssic problem of Functionl Anlysis is to determine when the limit of sequence of integrble functions is integrble nd the integrl of this limit

2 2452 Jun Alberto Escmill et l. function is the limit of the integrls; i.e., if (f n ) is sequence of integrble functions on A, which converges to function f on A, then it is of interest to know when the function f is integrble on A nd lim f n = f. (1) n A A The solution to this problem depend principlly of three fctors: the kind of convergence tht is being consider, the domin under which the functions re defined nd the kind of integrl tht is being used. For the cse of the Riemnn integrl (see Definition 2.4), it is known fct tht if (f n ) is sequence of Riemnn integrble functions on compct intervl [, b], which converges punctully to function f on [, b], not necessrily it holds tht the function f is Riemnn integrble on [, b], see [1]; even more, lthough the function f be Riemnn integrble on [, b], we cnnot ssure, in generl, tht the vlue of the Riemnn integrl of f on [, b] be the limit of the integrls of the functions belonging to the sequence (f n ), see [1]. However, it is known fct tht if we chnge the punctul convergence by the uniform convergence, then the limit function is Riemnn integrble nd the vlue of its integrl is equl to the limit of the integrls; i.e., (Uniform Convergence Theorem) If (f n ) is sequence of Riemnn integrble functions on compct intervl [, b], which converges uniformly to function f on [, b], then f is Riemnn integrble on [, b] nd the equlity (1) holds; where the intervl [, b] plys the role of subset A in (1). However, the bove convergence theorem is not vlid, in generl, for the improper Riemnn integrl (see Definition 2.5), since there re sequences of integrble functions in the sense improper of Riemnn, which converge uniformly to function such tht this limit function is not integrble in the sense improper of Riemnn; for exmple: Consider the sequence of functions (f n ) defined on the intervl [1, ) of the following wy f n (x) = { 1/x if 1 x n, n/x 2 if x n. (2) On this wy, it holds tht every f n is integrble in the sense improper of Riemnn on [1, ) nd, lso, the sequence (f n ) converges uniformly to the function f(x) = 1/x on [1, ); however, this limit function is not integrble in the sense improper of Riemnn on [1, ). Indeed, lthough the limit function be integrble in the sense improper of Riemnn, not necessrily it holds tht the vlue of this improper integrl be the limit of the improper integrls, see [1]. At lest, we cn follow two wys for to resolve the bove problem:

3 A convergence theorem for the improper Riemnn integrl 2453 Adding conditions to the sequence of functions or chnging the kind of convergence. Concerning to dd conditions to the sequence of functions, it is known fct, in prticulr, tht if to sequence (f n ) of integrble rel-vlued functions in the sense improper of Riemnn on A, which converges uniformly to function f on A, we dd the condition tht the sequence is dominted by function g, it mens, f n (x) g(x), for every x A nd for every n N, where g is integrble in the sense improper of Riemnn on A, then the limit function f is integrble in the sense improper of Riemnn on A nd the equlity (1) holds. On the other hnd, it is not common to find works tht del with the topic of to chnge the kind of convergence, even for the rel cse, i.e. when X = R; therefore, in this pper, we will prove tht if we chnge the uniform convergence by vrint of convergence defined by Moore in [4], then the limit function is integrble in the sense improper of Riemnn nd the vlue of this improper integrl is the limit of the improper integrls (see Theorem 3.2); moreover, the restriction of this result to compct intervls grees with the Uniform Convergence Theorem. 2 Preliminries For the reder s convenience, in this section, we will estblish the concepts tht will be used in the principl section of this pper. Definition 2.1 It is sid tht the sequence (f n ) converges uniformly to f on A if, nd only if, it holds the following: For every ε > 0 there exists n m N such tht if n m, then f n (x) f(x) < ε, for every x A. In the following definition nd, in some prts of the successive, the following nottion will be used: C + (A) = { ε : A R ε is continuous on A nd ε(x) > 0, for every x A}; if the subset A hs the form [, ), then we will write C + [, ) insted of C + (A).

4 2454 Jun Alberto Escmill et l. Definition 2.2 It is sid tht the sequence (f n ) converges in the sense of Moore to f on A if, nd only if, it holds the following: For every ε C + (A) there exists n m N such tht if n m, then f n (x) f(x) < ε(x), for every x A. Note 1. The convergence estblished in Definition 2.2 we hve clled it convergence in the sense of Moore, becuse it is bsed in the ides of convergence defined by Moore in [4] for the study of some integrble equtions; for dditionl informtion to go to Section of Commentries. Every constnt function defined on A, where the constnt is positive rel number, is positive nd continuous function; thus, if (f n ) is sequence of functions, which converges in the sense of Moore to function f on A, then this sequence lso converges uniformly to f on A; lthough the converse, in generl, is not true. However, if the sequence (f n ) converges uniformly to f on compct subset, then (f n ) lso converges to f in the sense of Moore; thus, on compct subsets, the uniform convergence nd the convergence in the sense of Moore re equivlents. Proposition 2.3 [4] If A is compct subset nd the sequence (f n ) converges uniformly to f on A, then the sequence (f n ) converges in the sense of Moore to f on A. Since the convergence in the sense of Moore implies the uniform convergence, we cn estblish the following: If (f n ) is sequence of integrble functions on [, b], which converges in the sense of Moore to f on [, b], then the function f is integrble on [, b] nd the equlity (1) holds, where [, b] plys the role of A in (1). The bove convergence theorem, which is bsed in terms of the convergence in the sense of Moore, it holds for processes of integrtion defined on compct intervls nd tht hold the uniform convergence theorem; like the integrl of Riemnn, McShne, Henstock-Kurzweil, nd so forth. Reciproclly, on the bses of Proposition 2.3, it holds tht the uniform convergence theorem implies the bove convergence theorem. We re going to close this section with the definition of the improper Riemnn integrl; but before, we give the definition of the Riemnn integrl for functions tking vlues in Bnch spce. Definition 2.4 Let f : [, b] R X be function. It is sid tht f is Riemnn integrble on [, b] if, nd only if, there exists z X with

5 A convergence theorem for the improper Riemnn integrl 2455 the following property: For every ε > 0 there exists δ > 0 such tht if P = {t 0 = < < t n = b} is prtition of intervl [, b], which P δ (with P = mx{t i t i 1 i = 1,, n}), then n (t i t i 1 )f(s i ) z < ε; i=1 where s i [t i 1, t i ], for every i = 1,, n. Note 2. Although for every Bnch spce X, the collection of ll Riemnn integrble functions f : [, b] X is liner spce nd the Riemnn integrl is linel opertor over it, the Riemnn integrl, in generl: it is not n bsolute integrl, the collection of ll discontinuity points of Riemnn integrble function not necessrily is of Lebesgue mesure zero nd, moreover, not ll Riemnn integrble functions re Bochner mesurble (in contrst with the rel cse; i.e. when X = R); see [9]. Definition 2.5 Let R. It is sid tht f : [, ) X is integrble in the sense improper of Riemnn on [, ) if, nd only if, for every b > it holds tht f is Riemnn integrble on [, b] nd, moreover, this limit lim b b f exists. In such cse, b f = lim f. b Note 3. On similr wy, it be define the improper Riemnn integrl for functions defined on n intervl of the form (, ]; for the cse (, ), it is defined in terms of the improper Riemnn integrl on the intervls (, ] nd [, ). 3 Min Results In prticulr, we re interested in the improper Riemnn integrl for intervls not bounded: [, ), (, ] nd (, ). Thus, in the following result, we estblish crcteriztion of the convergence in the sense of Moore on n intervl [, ); in like mnner, it cn be prove for the others kinds of intervls for which we re interested. Note 4. In Lemm 3.1, the expression fn f 0 on [M, ) mens f n (x) f(x) = 0, for every x [M, ); where 0 denote the null vector of Bnch spce X.

6 2456 Jun Alberto Escmill et l. Lemm 3.1 Let R. The sequence (f n ) converges in the sense of Moore to f on [, ) if, nd only if, there exists n M > such tht (f n ) converges uniformly to f on [, M] nd f n f 0 on [M, ), except for finite number of indexes n. Proof. = ) Suppose the contrry; i.e., we suppose tht the sequence (f n ) converges in the sense of Moore to the function f on [, ) nd, moreover, for every M > it holds tht I) (f n ) not converges uniformly to f on [, M] or II) f n f 0 on [M, ), for n infinity number of indexes n; however, ccording to the prgrph tht ppers fter of Note 1, it holds tht the condition I) cnnot be true. Thus, let M 1 = + 1; therefore, ccording to condition II), there exists n 1 N such tht f n1 f 0 on [M 1, ), i.e. there exists x with the property tht f n1 (x 1 ) f(x 1 ) 0. Now, let M 2 = x 1 + 1; therefore, ccording to condition II), there exists n 2 N with n 2 > n 1 such tht f n2 f 0 on [M 2, ), i.e. there exists x 2 > x 1 with the property tht f n2 (x 2 ) f(x 2 ) 0. Then, mking this procedure of recursive mnner, we cn mke two sequences (f nk ) nd (x k ) with the following properties: x k+1 > x k, for every k N; (x k ) converges to nd f nk (x k ) f(x k ) 0, for every k N. Now, we set c k = f nk (x k ) f(x k ) /2 nd ε : [, ) R of the following wy c 1 if x [, x 1 ], ε(x) = c k if x = x k, k N, (3) c k+1 c k x k+1 x k (x x k ) + c k if x [x k, x k+1 ], k N. On this wy, it holds tht ε C + [, ) nd, therefore, there exists m N such tht if n m, then f n (x) f(x) < ε(x), for every x [, ). Thus, for every k N such tht n k > m, it holds, in prticulr, tht f nk (x k ) f(x k ) < ε(x k ); which is contrdiction. =) Now, suppose tht there exists M > such tht I) (f n ) converges uniformly to f on [, M] nd II) f n f 0 on [M, ),

7 A convergence theorem for the improper Riemnn integrl 2457 except for finite number of indexes n; thus, let {n j } k j=1 be the collection of ll indexes n such tht not hold with II). Then, ccording to I) nd on the bses of Proposition 2.3, it holds tht (f n ) converges in the sense of Moore to f on [, M]; therefore, if ε C + [, ) there exists m N such tht if n m, then f n (x) f(x) < ε(x), for every x [, M]. Thus, if n N = mx{n, n 1,, n k } nd x [, ), then f n (x) f(x) < ε(x); which implies tht (f n ) converges in the sense of Moore to f on [, ). We shll now estblish, on the bses of Lemm 3.1, the following convergence theorem for the improper Riemnn integrl of functions tking vlues in Bnch spce. Theorem 3.2 Let R. If (f n ) is sequence of integrble functions in the sense improper of Riemnn on [, ), which converges in the sense of Moore to function f on [, ), then f is integrble in the sense improper of Riemnn on [, ) nd lim f n = f. (4) n Proof. Since the sequence (f n ) converges in the sense of Moore to the function f on [, ), it holds tht, on the bses of Lemm 3.1, there exists M > such tht: I) (f n ) converges uniformly to f on [, M] nd II) f n f 0 on [M, ), except for finite number of indexes n. Thus, ccording to I) nd II) nd, on the bses of Uniform Convergence Theorem for the Riemnn integrl, it holds tht f is Riemnn integrble on [, M] nd [M, b], for every b > M nd, moreover, the lim b b f exists; then, since f = M it holds tht f is integrble in the sense improper of Riemnn on [, ). f + M f,

8 2458 Jun Alberto Escmill et l. Now, let see tht the equlity (4) holds; in fct, observe tht f n f = = M M f n + f n f f M M ( M ) M ( ) f n f + f n f M M. On the other hnd, since the improper Riemnn integrls M f n nd M f exist nd s M f n M f = M (f n f) = 0, it holds tht f n f = M M f n f. (5) Therefore, ccording to equlity (5) nd on the bses of Uniform Convergence Theorem for the Riemnn integrl, it holds tht lim f n = f. n 4 Commentries On similr wy like with the uniform convergence, the convergence in the sense of Moore is equivlent to convergence generted by topology; in fct, if for every f X A = {f : A X f is function } we consider the collection {B(f, ε) ε C + (A)}, (6) where B(f, ε) = {g X A : g(x) f(x) < ε(x), for ll x A}, then this collection crete neighborhood bse of f; thus, the convergence generted by the topology induced by this collection of neighborhoods coincide with the convergence in the sense of Moore defined in X A. The bove topology, in contrst with the uniform topology, is not metrizble; however, this topology equips X A of structure of topologicl group, but not of topologicl vector spce, becuse the multipliction of vector by sclr is not continuous function. The ide of defining topologies on function spces by using functions ε rther thn constnts ε to specify the neighborhoods, goes bck t lest to Moore [4]; but it hs lrgely been restricted to spces of continuous functions nd considering continuous nd positive ε, s with the m-topology. It mens, if we consider the spce C(X) (where C(X) is the spce of ll continuous relvlued functions defined on X) insted of X A, then the topology generted by the collection of ll bses of neighborhoods defined s in (6), it is clled the m-topology; for reders interested in to know some topologicl properties

9 A convergence theorem for the improper Riemnn integrl 2459 of the m-topology, see [10]. Another ppliction, of quite different nture, of this ide to use functions ε rther thn constnts ε is to integrtion theory; see [3], [7] nd [8]. Finlly, we would like to mention tht the convergence in the sense of Moore cn be chnged nd dpted to severl frmeworks; in prticulr, if insted of to consider continuous nd positive ε, we consider tht these ε hve nother properties, then we obtin other useful topologies. For instnce, if A = R, X = R, nd, for every f R R, consider the neighborhood bse of f creted by the collection {B(f, ε) ε L + (A)}, where L + (A) = { ε R R : ε is Lebesgue mesurble nd ε(x) > 0, for ll x A}, then the topology generted by this neighborhood collection on R R is clled the topology of close pproximtion. In [2] be nlyze the subspce L = {f : R R f is Lebesgue mesurble} of R R, in the reltive topology induced by the topology of close pproximtion; the motivtion of to nlyze this topology on the spce L comes from [5] nd [6], where it is shown tht pproximtion in this close sense hs some useful properties. References [1] Robert G. Brtle, The Elements of Rel Anlysis, John Wiley nd Sons, Inc., United Sttes of Americ, [2] E.K. vn Douwen nd A.H. Stone, The topology of close pproximtion, Topology nd its Applictions, 35(1990), [3] R. Henstock, A Riemnn-type integrl of Lebesgue power, Cnd. J. Mth., 20, (1968), [4] E. H. Moore, On the fundtions of the theory of liner integrl equtions, Bull. Amer. Mth., 46 ( ), ,. [5] D. Mhrm nd A.H. Stone, One-to-one functions nd problem on subfields, in: Mesure Theory Oberwolfch 1979, Lectures Notes in Mthemtics 794 (Springer, Berlin, 1980) pp [6] D. Mhrm nd A.H. Stone, Expressing mesurble functions by one-one ones, Adv. in Mth., 46(1982), [7] W. Pfeffer, The divergence theorem, Trns. Amer. Mth. Soc., 295 (1986),

10 2460 Jun Alberto Escmill et l. [8] W. Pfeffer, The multidimensionl fundmentl theorem of clculus, J. Austrl. Mth. Soc., 43(1987), [9] Jos R. Ruiz, Intregrles Vectoriles de Riemnn y McShne, Underground Thesis, Universidd de Murci, Spin, [10] M. G. Rggi, L m-topolog en Espcios de Funciones, Mster Thesis, Benemérit Universidd Autónom de Puebl, Mexico, Received: September 7, 2014; Published: October 28, 2014