Some Local Uniform Convergences and Its. Applications in Integral Theory

Transcription

1 teratioal Joural of Mathematical Aalysis Vol. 1, 018, o. 1, HKAR Ltd, htts://doi.org/ /ijma Some Local Uiform Covergeces ad ts Alicatios i tegral Theory Doris Doda * ad Agro Tato ** * Wisdom Uiversity, Albaia ** Tiraa Polytechic Uiversity, Albaia Coyright 018 Doris Doda ad Agro Tato. This article is distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited. Abstract this aer we roose some roositios i referece to the otio of exhaustiveess which alies for families ad sequeces of fuctios. This ew otio give for the first time from Gregoriades ad Paaastassiou has a good rogress ad i mathematical commuity it is followed by may aers. We are cocered i study these families i framework of exhaustiveess i order to aly to measure theory ad Hestock-Kurzweil itegratio. We ivestigate the roerties of oe well kow covergece, local uiformly covergece or short "-covergece" ad comare it with other covergeces. Keywords: Exhaustiveess, alha-covergece, exhaustive sequece, equicotiuity, locally uiformly covergece or -covergece, Hestock-Kurzweil itegral troductio The otio of α-covergece (other-vise, cotiuous covergece or steige Kovergez ) has bee kow by the begiig of the 0th cetury (see [4, 5]). Aroud 1950s Stoilov [9], Ares, Kelley ad others [,8] came u with some results which characterize α-covergece ad are very helful for our aer. Also this tye of covergece was cosidered i coectio with some other tyes of covergece i [3]. The covergece which we are cocered is the locally uiformly covergece (or short -covergece) ad strog locally covergece (short *- covergece) i order some alicatio o itegral theory.

2 63 Doris Doda ad Agro Tato the first sectio, we recall some imortat cocets ad roositios about - covergece ad i articularly the cocet of exhaustive sequeces. the secod sectio, we start with defiitio of locally uiformly covergece of sequece of fuctios. We recall the well kow defiitio of this covergece, that is: Let f : X Y be the sequece of toological (metric) sace to metric sace Y (or uiform). The this sequece is locally uiformly coverget if for every oit x X there is a eighborhood Ux such that f is uiformly o Ux. t seems i may articles that this cocet stoed here ad oe ofte relates it with locally comact covergece. That is: f is locally comact coverget if f is uiformly coverget i every comact subsets KX. the framework of the exhaustiveess [4], there exist the oortuities to use ad develoed further these otios. this sectio, we highlight some roofs for the relatio of locally uiformly covergece with oit-wise covergece, uiform covergece ad locally comact uiform covergece. The third sectio, we devote the relatio of local uiform covergece with - covergece ad show that this covergece which is weaker that - covergece. We have substitute it i some kow roositios. Oe secial care take ad the strog locally uiform covergece, *-covergece, which is a little strog that -covergece i some cases. the fourth sectio, we give alicatios of -covergece about some roerties of equi-cotiuity o the Hestock- Kurzweil itegratio which are extesively treated to Schwabik[8]. 1. Prelimiaries Let us begi with some commets o otatio. With X ad Y we mea metric saces, uless stated otherwise. f it is ot metioed exlicitly the symbol d stads for the metric o X ad the symbol for the metric o Y. f x is a member of X ad δ is a ositive umber, with S(x, δ) we mea the (oe) ball of radius δ, i.e. S(x, δ) = {y X/d(x, y) < δ}. Also, if X ad Y are metric saces as usual we deote with C(X,Y) the set of all cotiuous fuctios from X to Y. We ow reset the ew otio of exhaustiveess [3] which is close to the otio of equi-cotiuity. Defiitio 1.1. Let (X, d), (Y,) be metric saces, x X, F be a family of fuctios from X to Y ad f : X Y, N.

3 Some local uiform covergeces ad its alicatios i itegral theory 633 (1) f F is ifiite, we call the family F exhaustive at x if for each ε > 0 there exists δ > 0 ad A a fiite subset of F such that: for each y S(x, δ) ad for each f F\A we have that (f(, f(y)) < ε () case where F is fiite we defie F to be exhaustive at x if each member of F is cotiuous fuctio at X. (3) F is called exhaustive if F is exhaustive at every xx. (4) The sequece (f ) N is called exhaustive at x if for all ε > 0 there exist δ > 0 ad 0 N such that for all y S(x, δ) ad all 0 we have that ( f (y), f () < ε (5) The sequece (f ) N is called exhaustive if it is exhaustive at every x X. Remarks 1..[4] Notice that i the most iterestig case where (f ) N is a sequece of fuctios for which f f m for m,the the family F = {f N} is exhaustive at some x 0 X if ad oly if the sequece (f ) N is exhaustive at x 0. A equi-cotiuous family is a exhaustive family such that for each ε > 0 the fiite set A i Defiitio 1.1 ca be take to be the emty set. So equi-cotiuity imlies exhaustiveess. Proositio 1.3[4]. Let (X, d), (Y,) be metric saces, x X, F a family of fuctios from X to Y ad f : X Y, N. (1) F is equi-cotiuous at x if ad oly if F is exhaustive at x ad for each f F, f is cotiuous at x. () The family {f N} is equi-cotiuous at x if ad oly if the sequece (f ) N is exhaustive at x ad each f is cotiuous at x. Followig this imortat article of Gregoriades ad Paaastassiou [4] we see that recedig roositio suggests that there exists a exhaustive sequece (similarly, family) which cotais o cotiuous fuctios. deed this haes as we ca see i the followig examle. Examle 1.4. For N defie f : R R such that f ( = 1, for x 0, f ( = 1 x > 0. Of course o f is cotiuous at 0. We claim that the sequece (f ) N is exhaustive at 0. Let ε >0, the there exists a iteger 0 > 1 ε such that for δ = 1, for all y ( 1, 1) ad for all 0 we have that f (y) f (0) 1 < ε.

4 634 Doris Doda ad Agro Tato Defiitio 1.5. Let f, f, N be fuctios from X to Y. The sequece (f ) N α- coverges to f if for each x X ad for each sequece (x ) N of oits of X covergig to x, the sequece (f (x )) N coverges to f (. α We shall write f f to deote that (f ) N α-coverges to f. Also we will kee the aalogous otatio about oit-wise ad uiform covergece, i.e., we will deote w u them with f f ad f f resectively. Remarks 1.6. ([4]) (1) t is obvious that α-covergece is stroger tha oit-wise covergece. () The usual covergeces such as oit-wise ad uiform do ot require a toology for the domai sace. However a toology is eeded for α-covergece. (3) Take f: R R ay o-cotiuous fuctio ad x x such that the sequece (f(x )) N does ot coverge to f (. f we ut f f for all N, we see that (f ) N does ot α-coverge to f although the sequece (f ) N coverges uiformly to f. (4) For all N defie f : (0,1] R such that f = 1 x, for x 1 ad f ( = 0,for x > 1. The we ca see that the sequece (f ) N α-coverges to zero fuctio but does ot coverge uiformly. The ext roositio is due to Stoilov [9] excet the last assertio ad describes some iterestig results about α-covergece. Proositio 1.7. Let (X, d), (Y,) be metric saces ad fuctios f, f, N, from X to Y. (1) f the sequece (f ) α-coverges to f, the f is cotiuous. () The sequece (f ) α-coverges to f if ad oly if f is cotiuous ad (f ) N coverges to f uiformly o every comact subset of X. articular: (3) f (f ) coverges to f uiformly ad f is cotiuous, the (f ) N α- coverges to f. Ad also: (4) f X is comact ad (f ) α-coverges to f, the (f ) N coverges to f uiformly. The followig result is due to Holá Šalát [8]. (5) A metric sace X is comact if ad oly if for all fuctios f, f, N, from X to Y, if (f ) N α-coverges to f, the (f ) N coverges to f uiformly.

5 Some local uiform covergeces ad its alicatios i itegral theory 635. Local uiformly covergece ad exhaustive sequeces Defiitio.1: Let (X,d), (Y,) be metric saces, xx ad f, f : X Y. Fuctio f( is a locally uiformly limit or δ limit of the sequece (f ) N if for each ε > 0, there exist 0 (, N ad δ > 0 that for each 0, ad y S(x, δ) we have (f (y), f(y)) < ε. So, we say that (f ) N coverges locally uiformly to f( or short is δ coverget to f(. This covergece i uiformly o oe oe eighborhood of a oit. So the fuctio f(=x if we see oly [0,1[ is - coverget i this iterval but as it is kow this sequece is ot uiformly coverget. Proositio.: Let f X Y be a sequece that is oit-wise coverget to a fuctio f: XY for each x X.The the sequece f δ coverges to f for each x X. The roof is clear by the defiitio (.1). The followig statemets rereset some the mai feature of local uiform covergece which defie its ositio i resect with oit-wise ad uiform covergece. Proositio.3. Let (X,d), (Y,) be metric saces, xx ad f, f : X Y. (1) f the sequece f () x is exhaustive ad -coverges to the fuctio f(. The the fuctio f( is cotiuous i this oit. () f the sequece f () x cosistig cotiuous fuctios ad -coverges to the fuctio f(. The the fuctio f( is cotiuous i this oit. Proof. (1) We call the defiitio of -covergece ad exhaustiveess of the sequece of these fuctios. First, by virtue of -covergece for every >0 ad 3 ad xx there exists the umber 0 (ε, ad 1>0 such that for (, ad 0 y S( x, 1) we have ( f( y), f ( y)). Also, from the roerty of exhaustiveess for this >0 ad 3

6 636 Doris Doda ad Agro Tato xx there exists a umber 1 (ε, ad 0 such that for (, ad 1 y S( x, ) derive ( f( y), f( ). We write 3 (, max{ (,, (, )} ad =mi{1,} the for (, ad 0 1 y S( x, ) we get the iequality ( f (, f ( y) ( f (, f( ) ( f(, f( y)) ( f( y), f ( y)) ad rove the cotiuous of f at x. () The secod assertio is roved similarly from the iterretatio of same equatio. Corollary.4. f the sequece f : X Y is δ coverget to the fuctio f( at a oit x, ad X is comlete the the sequece (f ) is Cauchy. Proof. By the δ covergece we write for every ε > 0, there exists δ 1 > 0 ad 0,such that for 0 ad y S(x, δ 1 ),we have (f (y), f(y)) < ε,also m 0 ad δ > 0,if y S(x, δ ) we have (f m (y), f(y)) < ε. As a coclusio for m, 0 ad δ = mi(δ 1, δ ) we take ( fm( y), f( y)) ( f( y), f ( y)) ( f ( y), f( y)) Proositio.5. Let (X,d), (Y,) be metric saces, xx ad f, f : X Y. The sace X is comact ad the sequece sequece f coverges uiformly to this fuctio. f -coverges to the fuctio f. The the Proof. By the defiitio of -covergece for every >0 (we fix oe) ad every xx exist the umber ω (ε, x ω )ε ad >0 such that for (, x ) ad y S( x, ) we have ( f( y), f ( y)). Let we cosider oe ifiite cover of X by the balls{ Sx (, )}. By virtue of defiitio of comact sace i metric sace there exists a fiite cover m X S ( x, ) k1 k k f we cosider ow the fiite sequece of the umbers 1 {,..., } ad 0 the maximum of this sequece. Hece for 0 ad yx there exists o ball Sx (, ) i such that if y S( x, ) i i the ( f ( ), ( )) i y f y. m

7 Some local uiform covergeces ad its alicatios i itegral theory 637 Corollary.6. The sequece locally uiformly coverget is locally uiformly comact coverget. f the X is locally comact the coverse is true. 3. Relatio betwee, a ad *-covergece with -covergece Amog others, we have also foud a defiitio "of the tye," for -covergece with little restrictios ad we ame it strog locally uiformly coverget. Defiitio 3.1. Let (X,d), (Y,) be metric saces, xx ad f, f : X Y.We say that sequece (f () is strog locally uiformly coverget (or short a - coverget) to f( if for every >0 ad xx, there exists 0 (ε, ad >0 such that for (, ad y S( x, ) 0 we have ( f ( y), f ( ). We ca rofit some roerties of -covergece for a -covergece with other roofs. Proositio 3.. Let (X,d), (Y,) be metric saces, xx ad f, f : X Y ad the sequece f : X Y is a - coverget to the fuctio f(. The a) Sequece (f () N is exhaustive, b) f( is cotiuous at xx Proof. a) By the defiitio of a - covergece for every >0 ad 0, xx, there exists 0 (ε, ad >0 such that for (, ad y S( x, ) we 0 have ( f( y), f ( ). Accordig this 0 (, ad y S( x, ) we get ( f( y), f( ) ( f( y), f( ) ( f (, f( ). b) f i above roof, we take 3 i lace of >0 ad ay xx, there exists 1 (ε, ad 1>0 such that for (, ad y S(x,) we have ( ( ), ( )) 1 f y f x. Also there exists (ε, ad >0 such that for (, ( f( y), f ( ) ad ( f(, f ( ). 3 3

8 638 Doris Doda ad Agro Tato The secod iequality shows that from a - covergece comes oit-wise covergece. Hece for every >0 ad, there exists (, y) 3 such that for (, y) we get ( ( ), ( )) f y f y. So for x,ys(x,) where = mi(1,) if 3 we take a 4 max( 1,, 3 ) for (, 3 we obtai ( f (, f ( y)) ( f (, f ( ) ( f (, f ( y)) ( f ( y), f ( y)) t looks like the two cocets -covergece ad a - covergece are equivalet. reality, cocet of -covergece used by Kelley [7] i ets o toology is 1 cosidered that two covergeces x x ad fm( f( are quite differet. f we limit ourselves oly to the case whe two idexes ad m of the first ad the secod covergece are comarable o the we have a stroger covergece, that is: we say that the sequece f is *-coverget o X if for every >0 ad xx there exists a umber 1 such that for >1 the sequece x x ad there exists a umber such that for >, f ( x ) f (. Examle 3.3. Let we show a case that from - covergece do't derives *- covergece. ( i k 1)( i k ) We take ( which go to zero i oe secial way whe i that is that if we go diagoals i the below ifiite table. For examle, the third diagoal is a31, a, a13 the sum of idexes is 4. So, we have costruct a order 1 with the air of idexes i such way: ( i, k) ( i', k ') if i k i ' k '. f as the sequece x i X we take x 1 i k for the elemets i oe diagoal of the table bellow, for examle i+k=4 we have that x4=(a31) x5=(a) x6 =(a13). These x have the value 1/4 for =4,5,6. t easy to see that i k i case where the x0. Le we ut f ( x ) x go to zero by order 1. Let we come back to the ifiite table a a a a a a 1 3 a a a

9 Some local uiform covergeces ad its alicatios i itegral theory 639 Now we ca choose aother order. By this order set, idex take the values by "Determiat rule", ad, for examle, we have a art of sequece a11 a1 a a1... which is corresoded the sequece x1,x, x3,x4,...of order 1. t is clear that ad i this case i k but if we fix, we fid o =100=k+i=50+50 such that for >100 f( x ) 0 x but the term x x 1 ad x>x+1 because (i,k) (i,k-1) = 50+49< 0 that cotradicts defiitio 3.1. Proositio 3.4. Let (X,d), (Y,) be metric saces, xx ad f, f : X Y. The followig are equivalet: 1) Let be >0 ad xx, there exists 0 (ε, ad >0 such that for (, 0 ad y S( x, ) we have ( f ( y), f ( ), ) The sequece (f () *-coverges to f(. Proof. (1) (). Let us cosider the covergece x x, that is, for every 1>0, there exists 1 (δ 1 ) such that for 1( ) d( x, 1 or x S( x, 1). t follows that for every >0 there exists <1 ad max( 0, 1 ) such that whe x S( x, ) d( f (, f ( ). t is roved that f *-coverges to f(. ()(1).Let us assume that f () x is ot coverget accordig first statemet. Hece, there exists oe 0>0 such that for xx ad, (ε 0, ad >0 such that for ( 0, ad ys( x, ) ( f(, f ( ) 0. By the secod statemet whe x xthere exists oe >0 such that ( ) (, ) 1 d x x but f() x do't coverges to f( accordig Defiitio 3.1. That cotradicts (). Proositio 3.5.: Let f, f, N be fuctios from X to Y. f the sequece (f ) is *-coverget ad exhaustive the it is δ-coverget. Proof. Sice f is *-coverget for each ε > 0 ad for ε,there is δ 1 ad 1 1 such that for 1 ad x S( x, ) it follows that there exists such that ( f(, f ( ).

10 640 Doris Doda ad Agro Tato Give that (f ) is exhaustive for each ε > 0, there exists δ < δ 1, 0 that for 0 ad y S (x, δ ) we ca have the relatio (f (y), f() < ε for = max( 1, 0 ) ad δ < δ 1. We cosider the followig iequalities for y S(x, δ), x S(x, δ) f y, f y f y, f x f x, f x because of x S(y, δ). Examle 3.6 The followig sequece is -coverget to the set ]1-, 1+[ but o -covergece or uiform covergece: x for 1- x<1 1 x f( x for 1<x<1+ 1 x This sequece coverges oit-wise ad locally uiformly to the fuctio 1 for x=1 f( 0 otherwise but if the sequece x 1, x]1-,1] the f( - f(1) 0. Proositio 3.7. Let hold the coditios i above Proositios. The sequece f () x -coverges to fuctio f() x if ad oly if f() x is -coverget ad f() x is cotiuous fuctio. Proof. Necessary coditio is evidet because from -covergece derives acovergece ad from this -covergece (Proositio 3.5). Let we see the iverse statemet. By the defiitio of covergece for every >0 ad 0, xx, there exists 1 (ε, ad 1>0 such that for (, ad 0 y S( x, 1) we have ( f( y), f ( y)), also from the cotiuous of f() x for this there exists a >0 such that ( f (, f ( y). f we write =mi{1,} tha for (, ad y S( x, ) 0 we have ( f( y), f ( ) ( f( y), f ( y)) ( f ( y), f ( )

11 Some local uiform covergeces ad its alicatios i itegral theory Alicatio of exhaustive fuctios o Hestock-Kurzweil itegral The defiitio give o the metric sace we ca recostruct ad for the case of the ormed sace or Baach sace. Defiitio 4.1. a) Let f : SX, where S ad X are ormed sace the sequece (f) is called exhaustive sequece at s S if for every >0 there is a >0 ad 0 such that for every yb(s,) we get f ( y) f (. b) Sequece (f(s)) has the -limit the fuctio f(s) for every ss if ad oly if for every >0 there is a >0 ad 0 such that for every yb(s,) we have f ( y) f ( y) Let S R m be a comact iterval ad m 1. Recall that for the defiitio of the Hestock-Kurzweil itegral we eed the cocets of K-artitio [8].The air (r, S) where r m ad S is a comact iterval by m is called a tagged iterval such that r is a tag of S. K-system {(r j, j ) j = 1,,.. } is a collectio of fiite tagged o-overlaig itervals where r j j ad S = j j=1 K-system is called K-artitio of the iterval S if is true the revious equality. Let : S ]0, + [ be the fuctio, that is called gauge i S. The tagged iterval (τ, S) is called refied if we have S B(τ, (τ)) where B(τ, (τ)) is a ball i m with the ceter i τ ad radius (τ). K-system or the K-artitio are refied if all the tagged itervals (τ j, j ),j=1,,, are refied i coectio with the gauge. Now, Let S is a measurable set with algebra ad robability measure, we write {,, } Defiitio 4..The fuctio f: X is Hestock-Kurzweil itegrable ad X is Hestock-Kurzweil itegral, if for every ε > 0 there is the gauge S ]0, + [ such that for every refied, K-artitio (t i, i ),i=1,,, of takes lace the iequality:

12 64 Doris Doda ad Agro Tato f(t i )μ( i ) J < ε i=1 The Hestock-Kurzweil itegral(or short (HK) we deote by the symbol J ( HK) f Defiitio 4.3.The fuctio f: X has the roerty S HK if for every ε > 0 there is the gauge i such that we take k i=1 l j=1 (1) f(t i ) f(s j ) X μ(j i L j ) < ε For every refied K-artitios {(t i, i ), i = 1,,, k} ad {(s j, l j ), j = 1,,, l} i. Proositio 4.4. f we have = {f: X} a set of exhaustive fuctios the those are S HK itegrated. Proof. By choosig the value of that both artitios ca be refied, for every ε > 0, there exists μ() 1 <, that for t i S(s i, 1 ) we will have i=1,,,. f(t i ) f(s i ) < ε μ() From which we rove the iequality (1). Defiitio 4.5: A collectio M of the fuctios f: X, is a comact iterval o R is called HK-equi-itegrable if every f M is Hestock-Kurzweil itegrable ad for ay ε > 0 there is a gauge such that for ay f M the iequality i 1 i i f ( t ) ( ) ( HK) f holds rovided {(t i, i ), i = 1,,, } is a -refied K-artitios of. ext roositio we try to substitute the requiremet of HK-equi-itegralility with δ a covergece. Proositio 4.6: Let the fuctios f be of the collectio of fuctio f k : X, HK-itegrable ad δ a coverget to f such that lim f ( t) f ( t). a k k The the fuctio f: X is HK itegrable ad

13 Some local uiform covergeces ad its alicatios i itegral theory 643 holds. lim( HK) f ( HK) f k Proof: Let is the gauge ad the K- artitio {(t i, i ), i = 1,,, } which is - fie of S for ay k N, from the itegrability of fk we write f ( t ) ( ) ( HK) f i1 k k i i k for every -fie K- artitio{(t i, i ), i = 1,,, } of. f the artitio {(t i, i ), i = 1,,, } is fixed, the a covergece leads that for every ti there exists i>0 ad k0 such that for k>k0 ad ris(ti,i)i yields f ( r ) f ( t ) k i i () Let we costruct a other K-artitio {(ri,i), i=1,,...,}. This ew artitio is also -fie ad holds. Therefore for k >k0. This gives for k,l >k0 the iequality i1 k i i i1 i i X f ( r ) ( ) f ( t ) ( ) f t i 1 i i HK f k X ( ) ( ) ( ) i1 i1 [ f ( t ) ( ) f ( r ) ( ) i i k i i f ( r ) ( ) ( HK) f i i k X ( HK) f ( HK) f 4 k l X which shows that the sequece ( HK) fk k of elemets of X is Cauchy sequece i the Baach sace X. The exists its limes JX: lim k f k J

14 644 Doris Doda ad Agro Tato Suose that {(t i, i ), i = 1,,, } is -fie K- artitio of. For k>n we obtai i1 i i X i1 i i k i i X f ( t ) ( ) J [ f ( t ) ( ) f ( r ) ( )] f ( r ) ( ) ( HK) f ( HK) f J 3 k i i k X k X i1 Ad it follows that f is Hestock-Kurzweil itegrable o ad k. lim( HK) f J ( HK) f k Refereces [1] R.F. Ares, A toology for saces of trasformatios, The A. of Math., 47 (1946), o. 3, htts://doi.org/10.307/ [] C. Carathéodory, Stetige kovergez ud ormale Familie vo Fuktioe, Mathematische Aale, 101 (199), o. 1, htts://doi.org/ /bf [3] R. Das, N. Paaastassiou, Some tyes of covergece of sequeces of real valued fuctios, Real Aal. Exchage, 8 (00/003), o., [4] V. Gregoriades, N. Paaastassiou, The otio of exhaustiveess ad Ascolitye theorems, Toology ad its Alicatios, 155 (008), o. 10, htts://doi.org/ /j.tool [5] H. Hah, Reelle Fuktioe, Chelsea, New York, [6] L. Holá, T. Šalát, Grah covergece, uiform, quasi-uiform ad cotiuous covergece ad some characterizatios of comactess, Acta Math. Uiv. Comeia., (1998), [7] J. Kelley, Geeral Toology, Sriger- Verlag, [8] S. Schwabik, Y. Guoju, Toics i Baach Sace tegratio, World Scietific, Sigaore, 005. htts://doi.org/10.114/

15 Some local uiform covergeces ad its alicatios i itegral theory 645 [9] S. Stoilov, Cotiuous covergece, Rev. Math. Pures Al., 4 (1959), Received: November 1, 018; Published: December 13, 018