A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction

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1 Ttr Mt. Mth. Publ. 44 (29), DOI: /v z t m Mthemticl Publictions A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES Miloslv Duchoň Peter Mličký ABSTRACT. We present Helly type theorem for sequences of functions with vlues in metric spces nd pply it to representtions of some mppings on the spce of continuous functions. A generliztion of the Riesz theorem is formulted nd proved. More concretely, representtion of certin mjored liner opertors on the spce of continuous functions into complete metric spce. 1. Introduction It is well known tht the following theorem is true [BDS], [N], [W]. Riesz representtion theorem. Every continuous liner functionl L on the set of continuous functions f defined on [, 1] hs the form Lf = f(s) dg(s) (R) with function g of bounded vrition on [, 1]. This theorem hs mny extensions nd generliztions with vrious proofs. One of the possible proofs is bsed on the Helly theorem [N] nd lso on the moment problem theorem. It cn be shown tht the problem of determining the generl continuous liner functionls on the set of continuous functions is equivlent to tht of determining the set of ll moment sequences. It is our purpose to extend this result for mjored liner opertors from continuous functions to Bnch spces. Helly s theorem hd been of some importnce long time bove ll in the probbility theory in connection with problem of moments of distributions. Recll tht in this connection, rel-vlued nondecresing functions f on the intervl [, b] of the rel line re considered nd tht the following fcts re true. 2 M t h e m t i c s Subject Clssifiction: 28B99, 44A6. K e y w o r d s: metric spce, vector spce, vector function, bounded vrition, mjored opertor. Supported by grnt gency Veg, 2/4137/7. 159

2 MILOSLAV DUCHOŇ PETERMALIČKÝ (1) (First Helly s theorem) Given uniformly bounded sequence (f n )ofrelvlued nondecresing functions, there exists subsequence (f nk )of(f n ) converging to rel-vlued nondecresing function f on [, b]. (2) (Second Helly s theorem) Given sequence (f n ) of rel-vlued nondecresing functions on [, b], converging to rel-vlued nondecresing function f, then, for every continuous function g on [, b], we hve lim g(t) df n (t) = g(t) df (t). More generlly there re true the following fcts [BDS], [Di], [N], [W]. (1) (First Helly s theorem ) Given the sequence (f n ) of complex-vlued functions of uniformly bounded vrition on [, b] such tht for some x [, b] the sequence ( f n (x ) ) is bounded then there exists subsequence (f nk )of (f n ) converging to some function f of bounded vrition on [, b]. (2) (Second Helly s theorem) Given sequence (f n ) of functions of uniformly bounded vrition on [, b], converging to some function f of bounded vrition, then, for every continuous function g on [, b], we hve lim g(t) df n (t) = g(t) df (t). 2. A Helly theorem in metric spces In this section we shll prove some fcts concerning Helly s theorem in the context of metric spces. Definition. Let D be subset of the rel line, (X, d) be metric spce nd h : D X be function. If the set of ll sums n d( h(t i 1 ),h(t i ) ),where (t i ) n i= is n incresing sequence elements of D, is bounded then g is sid to be function of bounded vrition on D. The corresponding lest upper bound is vrition of function h on set D. Proposition. Let D be dense subset of the intervl [, b], (X, d) be complete metric spce nd h : D X be function of bounded vrition on D. (i) For ny t (, b] (resp. t [, b)) thereexistslimith(t ) (resp. h(t+)) (ii) Function f :(, b] X (resp. g :[, b) X) defined by f(t) =h(t ) (resp. g(t) = h(t +)) re functions of bounded vrition on (, b] (resp. [, b)). 16

3 A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES (iii) f(t ) = g(t ) = h(t ) for ll t (, b] nd f(t+) = g(t+) = h(t+) for ll t [, b). (iv) Function f (resp. g) is continuous t the point t (, b) if nd only if h(t ) = h(t +). (v) For ll t (, b) except possibly countble set h(t ) = h(t +). Proof. Let (t i ) i= be n incresing (resp. decresing) sequence of elements ( of D. Since (X, d) is complete metric spce, it is sufficient to show tht h(ti ) ) is Cuchy sequence. To see this, note tht series i= d ( h(t i 1 ),h(t i ) ) is convergent nd d ( h(t n ),h(t m ) ) m d ( h(t i 1 ),h(t i ) ) i=n+1 d ( h(t i 1 ),h(t i ) ) for ny n<m. i=n+1 It proves (i). Let (t i ) n i= be n incresing sequence of elements of (, b]. Tke ε> nd sequence (s i ) n i= of D such tht s <t <s 1 <t 1 <...<t n 1 <s n <t n nd d ( f(t i ),h(s i ) ) < ε 2n.Then d ( f(t i 1 ),f(t i ) ) (d ( f(t i 1 ),h(s i 1 ) ) + d ( h(s i 1 ),h(s i ) ) + d ( h(s i ),f(t i ) )) d ( h(s i 1 ),h(s i ) ) + <ε+ d ( f(t i 1 ),h(s i 1 ) ) + d ( h(s i 1 ),h(s i ) ). d ( f(t i ),h(s i ) ) It mens tht f is of bounded vrition. Bounded vrition of g my be proved nlogously. It proves (ii). Now we prove f(t ) = h(t ) for t (, b]. Let (t i ) i= be n incresing sequence of elements of (, b] with lim t n = t. Tke sequence (s i ) i= of D such tht s <t <s 1 <t 1 <... < t n 1 <s n < 161

4 MILOSLAV DUCHOŇ PETERMALIČKÝ t n <... nd d ( f(t i ),h(s i ) ) < 1 n. Clerly Equlities nd f(t ) = lim f(t n) = lim h(s n)=h(t ). g(t ) = h(t ) for t (, b], f(t +)=h(t +) g(t +)=h(t + ) for t [, b) my be proved nlogously. It proves (iii). Prt (iv) follows from (iii). Let M be vrition of g on D nd n> be fixed. Assume tht there re m points t 1,...t m,wherem>nm, such tht inequlity d ( f(t i ),h(t i +) ) > 1 n is stisfied for ll i =1,...,m.Thereisε> such tht d ( f(t i ),h(t i +) ) > 2ε + 1 n for ll i =1,...,m. We my ssume tht (t i ) m is n incresing sequence. There re sequences (s i ) m nd (u i) m in D such tht s 1 <t 1 <u 1 <s 2 <t 2 <u 2 <...,t m 1 < u m 1 <s m <t m <u m, d ( f(t i ),h(s i ) ) <ε nd d ( h(t i +),h(u i ) ) <ε for ll i =1,...,m. Now m d ( h(s i ),h(u i ) ) m ( > d ( f(t i ),h(t i +) ) ) 2ε > m n >M wht is contrdiction. Therefore inequlity d ( h(t ),h(t +) ) > my be stisfied only for countbly mny t (, b). Helly theorem. Let (X, d) be complete metric spce nd (g n ) n N sequence of functions from [, b] into X such tht ) the set g n (x) is reltively compct for ny x [, b], b) the functions (g n ) n N hve uniformly bounded vritions. Then there exists subsequence of the sequence (g n ) n N converging pointwise in X tofunctiong :[, b] X of bounded vrition. Proof. Let D be countble set dense in [, b] ndb D. By the digonl procedure we obtin subsequence (h n ) n N of (g n ) n N converging for every 162

5 A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES t D to function h(t) which hs bounded vrition on D. Hence for every t [, b) there exists the limit h(t +).Put g(t) =h(t + ) for t [, b) nd g(b) =h(b). Let δ>begiven.weprovetht lim sup d ( h n (t),g(t) ) δ for every t [, b), except possible, finite number m 2M/δ, wherem is the upper bound of vritions of ll g n. So, ssume tht for some m>2m/δ there re points t 1 <t 2 <...<t m <bsuch tht lim sup d ( h n (t i ),g(t i ) ) >δ for ll i =1, 2,...,m. There is subsequence (h nk ) k=1 such tht d ( h nk (t i ),g(t i ) ) >δ for ll i =1, 2,...,m nd k =1, 2,... Since g(t i )=h(t i + ), there is sequence (u i ) m in D such tht d ( g(t i ),h(u i ) ) < δ for ll i =1, 2,...,m 4 nd t 1 <u 1 <t 2 <u 2 <...,t m 1 <u m 1 <t m <u m.letk be so lrge tht d ( h nk (u i ),h(u i ) ) δ/4, for ll i =1,...,m. Then m d ( h nk (t i ),h nk (u i ) ) > m (d ( h nk (t i ),g(t i ) ) d ( g(t i ),h(u i ) ) d ( h(u i ),h nk (u i ) )) m ( δ δ 4 δ ) 4 = mδ 2 >M. which is not possible. So, lim sup d ( h n (t),g(t) ) = for every t [, b), except possible, countble set A [, b). The lst ppliction of the digonl procedure gives sequence convergent to g(t) ona. Forsomet A it my be g(t) g(t). Therefore g hs to be redefined on A by g(t) = g(t). 163

6 MILOSLAV DUCHOŇ PETERMALIČKÝ Corollry ofhelly theorem. Let X be complete liner metric spce nd (g n ) n N sequence of functions from [, b] into X such tht ) the set {g n (x)} is reltively compct for ny x [, b], b) the functions (g n ) n N hve uniformly bounded vritions. Then there exists subsequence of the sequence (g n ) n N converging pointwise in X tofunctiong :[, b] X of bounded vrition. 3. Integrl representtion theorem for mjored liner opertor It is well known tht every continuous liner form on the spce of C(I), I =[, 1], is presentble in the form f f(t)dq(t), (1) where q is function of bounded vrition. From the preceding generliztion of Helly theorem we obtin the result giving the representtion of mjored liner mpping [Di]. First we recll the definition of mjored liner opertor. For ech subset A of [, b], let C ( [, b],a ) denote the spce of continuous functions on [, b] vnishing outside A. LetX be Bnch spce. If F : C ( [, b] ) X is liner mpping, define for ech A, F A =sup i F (f i ), where the supremum is over ll finite fmilies {f i } in C ( [, b],a ) with i f i χ A (t) for ll t in [, b]. If F : C ( [, b] ) X is liner mpping, then (see [Di, 19, pp. 38, 383]), the mpping F is clled mjored (lso dominted, [Di]), if F A < for ll A in B ( [, b] ). (M) 4. Riesz type theorem in some liner metric spces Let on the intervl [, b] function g of bounded vrition with vlues in Bnch X be given. This function my it possible to every continuous function 164

7 A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES f(x) on[, b] to ssocite the element of X of the form F (f) = f(x)dg(x). (!) The following properties re true. Let f 1,f 2,f C[, b]. Then F (f 1 + f 2 )=F(f 1 )+F(f 2 ), F (f) V b M(f), M(f) = mx f(x), Vb =Vr b (g). x [,b] Theorem (F. Riesz). Let on the set C[, b] the mjored nd compct liner mpping F (f) with vlues in Bnch spce X be given. Then there exists functiong(x) of bounded vrition with vlues X such tht for every function f(x) C[, b] we hve F (f) = () (b) f(x) dg(x). (2) P r o o f. It is enough to consider the cse =1,b= 1, becuse the generl cse cn be reduced to this cse by mens of liner trnsformtion of rgument. Put ( ) n ϕ n,k = x k (1 x) n k. k It is esy to see tht for every x it is true ϕ n,k (x) =1. k= Moreover, for x [, 1] every member of this sum in nonnegtive. Hence, if k = ±1, k=, 1,...,n, then k ϕ n,k 1. (3) k= Note tht the considered liner opertor F (f) is defined for continuous on [, 1] functions f(x). According to definition of compct liner opertor there exists compct subset C of X such tht F (f) M(f)C, or, equivlently, mpping the unit sphere in C[, 1] into C. From this nd (3) we obtin k F (ϕ n,k ) C. k= 165

8 MILOSLAV DUCHOŇ PETERMALIČKÝ Further from the mjority of opertor F we obtin F (f i ) F A <, (4) F [ϕ n,k ] sup k= i where the supremum is over ll finite fmilies f i C ( [, b],a ) with f i (t) χ A (t) for ll t [, b], for ll A B([, b]). Let us define the step function g n (x) to put g n () =, ( g n (x) =F [ϕ n, ] <x< 1 ), n ( 1 n x< 2 ), n g n (x) =F [ϕ n, + ϕ n,1 ] g n (x) =F g n (1) = F [ n 1 k= [ n k= ] ( n 1 ϕ n,k n ] ϕ n,k. ) x<1, By (4) the functions g n (x) nd their totl vritions re bounded with the one number. Moreover, becuse by ssumption F (f) is compct opertor on C[, 1] into X, it tkes unit sphere in C[, 1] into compct set C into X. Hence the set {g n (x)},n=1, 2,... is contined in C for ll x [, 1]. Hence on the bse of Helly principle of choice from the sequence { g n (x) } it is possible to choose the subsequence {g ni (x)} which converges in ech point of [, 1] to function of the bounded vrition. If f(x) is continuous function on [, 1], then on the bse of [N, Th. 3] it cn be shown tht from where where 166 f(x) dg n (x) = f k= f(x) dg n (x) =F [ B n (x) ], ( ) k F (ϕ n,k ), n

9 A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES B n (x) = f k= ( k n )( ) n x k (1 x) n k k is the Bernstein polynom for the function f(x). By the theorem of S. N. B e r n s t e i n [N, 5, Ch. IV] M(B n f), n, nd by the definition of continuous liner opertor we hve F (B n ) F (f) = F (B n f) KM(B n f). This mens tht F (B n ) F (f), from where lim f(x) dg n (x) =F (f). But if n, going through vlues n 1,n 2,..., then by Helly theorem (cf. lso [N, 7] nd [DD1]), we obtin lim f(x) dg n (x) = f(x) dg(x). Remrk. We hve derived here representtion theorem for mjored opertors using Helly theorem in liner metric spces. To prove it by mens of the Helly theorem from this pper we were ble to do it for compct mjored mppings. On the other hnd, our pproch is more constructive. In [Di] this is done for Bnch spces without such n ssumption. REFERENCES [BDS] BARTLE, R. G. DUNFORD, N. SCHWARTZ, J.: Wek compctness nd vector mesures, Cnd. J. Mth. 7 (1955), [DD1] DEBIEVE, C. DUCHOŇ, M. DUHOUX, M.: A Helly theorem in the setting of Bnch spces, Ttr Mt. Mth. Publ. 22 (21), [DD2] DEBIEVE, C. DUCHOŇ, M. DUHOUX, M.: A Helly s theorem in some Bnch lttices, Mth.J.ToymUniv.23 (2), [Di] DINCULEANU, N.: Vector Mesures. VEB, Berlin, [H] HAUSDORFF, F.: Momentprobleme für ein endliches Intervl, Mth. Z. 16 (1923),

10 MILOSLAV DUCHOŇ PETERMALIČKÝ [N] NATANSON, I. P.: Theory of Functions of Rel Vrible. Frederick Ungr Publishing Co., New York, [W] WIDDER, D. V.: The Lplce Trnsform. Princeton University Press, Princeton, Received October 16, 29 Miloslv Duchoň Mthemticl Institute Slovk Acdemy of Sciences Štefánikov 49 SK Brtislv SLOVAKIA E-mil: Peter Mličký Fculty of Nturl Sciences University of Mtej Bel Tjovského 4 SK BnskáBystric SLOVAKIA E-mil: mlicky@fpv.umb.sk 168