Uniform Distribution of Sequences and its interplay with Functional Analysis
aa r X i v : . [ m a t h . F A ] F e b Uniform Distribution of Sequences and its interplay withFunctional Analysis
S.K.Mercourakis and G.Vassiliadis
Abstract
In this paper we apply ideas from the theory of Uniform Distribution of sequences toFunctional Analysis and then drawing inspiration from the consequent results, we studyconcepts and results in Uniform Distribution itself. So let E be a Banach space. Then weprove:(a) If F is a bounded subset of E and x ∈ co ( F ) (= the closed convex hull of F ), thenthere is a sequence ( x n ) ⊆ F which is Ces`aro summable to x .(b) If E is separable, F ⊆ E ∗ bounded and f ∈ co w ∗ ( F ), then there is a sequence ( f n ) ⊆ F whose sequence of arithmetic means f + ··· + f N N , N ≥ ∗ -converges to f .By the aid of Krein-Milman’s theorem, both (a) and (b) have interesting implicationsfor closed, convex and bounded subsets Ω of E such that Ω = co ( ex Ω) and for weak ∗ compact and convex subsets of E ∗ . For instance, if we apply (b) to Ω = B M ( K ) , where K is a compact metric space, we get that for every signed Borel measure µ on K with k µ k ≤
1, there are sequences ( x n ) ⊆ K and ( ε n ) ⊆ {± } such that the sequence ε δ x + · · · + ε N δ x N N weak ∗ −→ µ. Assuming that k µ k = 1 we can prove in addition that ( x n ) can be chosen to be | µ | -u.d.,i.e. δ x + ··· + δ xN N weak ∗ −→ | µ | .By further expanding the previous ideas and results, we are able to generalize a classicaltheorem of Uniform Distribution which is valid for increasing functions ϕ : I = [0 , → R with ϕ (0) = 0 and ϕ (1) = 1, for functions ϕ of bounded variation on I with ϕ (0) = 0and total variation V ϕ = 1. So for every such ϕ , there are sequences ( x n ) ⊆ I and( ε n ) ⊆ {± } such that for each x ∈ I we havelim N →∞ N N X k =1 χ [0 ,x ) ( x k ) = υ ( x ) and lim N →∞ N N X k =1 ε k χ [0 ,x ) ( x k ) = ϕ ( x ) , where υ is the function of total variation of ϕ on I . Introduction
Our aim in this paper is twofold. We first study consequences of ideas coming from thetheory of Uniform Distribution of sequences [15] in Functional Analysis (section 1) and
Mathematics Subject Classification : Primary 46B09, 11K06; Secondary 40C05, 60B10 .
Key words and phrases : uniformly distributed sequence, Ces`aro summable sequence, function of boundedvariation. x in the convex hull co ( F ) of abounded subset of some Banach space, then for every N ∈ N , x can be approximated (anestimation of the approximation error is also given) by the arithmetic mean of N pointsof F . The second assertion says that, if x belongs to co ( F ), then there is a sequence ofpoints of F which is Ces`aro summable to x . A number of easy consequences of Theorem2 for a Banach space E are the following:A) If ( y n ) ⊆ E is any weakly null sequence, then there is a function ϕ : N → N , suchthat the sequence x n = y ϕ ( n ) , n ≥ E equal to the closed convex hull of itsextreme points ex Ω, then for every x ∈ Ω there is a sequence ( x n ) ⊆ ex Ω which isCes`aro summable to x (Prop.3). This result has, by the aid of Krein-Milman’s theorem,obvious implications for the unit ball of a Banach space which is reflexive or of the form C ( K ) (=the space of real continuous functions on K ), where K is any compact totalydisconnected space (Corollaries 1 and 2).C) A result analogous to Theorem 2(2) (with analogous proof) is valid for the dual( E ∗ , weak ∗ ) of a separable Banach space E . When F is a bounded subset of E ∗ and f belongs to co w ∗ ( F ), there is a sequence ( f n ) ⊆ F whose arithmetic means weak ∗ -converge to f (Prop.4).This result (again using Krein-Milman’s theorem) has obvious implications for aweak ∗ -compact and convex subset Ω of E ∗ , which generalize classical results (see Prop.5,Cor.3, Th.3 and Th.4). So Theorem 3 is the well known result stating that every µ ∈ Ω = P ( K ) ( K is any compact metric space) admits a u.d. sequence, but Theorem4 says something that seems to be new: for every signed measure µ ∈ Ω = B M ( K ) ,there is a sequence ( x n ) ⊆ K and a sequence of signs ( ε n ) ⊆ {± } , so that the sequence µ N = ε δ x + ··· + ε N δ xN N , N ≥
1, weak ∗ -converges to µ .The last result is our motivation for the second section of this paper. Drawinginspiration from Theorem 4, we extend the classical concept of uniformly distributedsequence defined for measures µ ∈ P ( K ), where K is a compact space (see Def. 1.1of [15]), to every signed measure µ ∈ M ( K ) with k µ k = 1. Thus we will say that asequence ( x n ) ⊆ K is µ -u.d. iff ( x n ) is | µ | -u.d. (in the classical sense) and if also thereis a sequence of signs ( ε n ) ⊆ {± } , so that the sequence µ N = ε δ x + ··· + ε N δ xN N , N ≥ ∗ -converges to µ (Def.1).Then (generalizing Theorem 3) we prove Theorem 5, which states that given a com-pact metric space K and µ ∈ M ( K ) with k µ k = 1, then µ admits a u.d. sequence( x n ) ⊆ K (in the sense of the aforementioned definition). So if f ∈ C ( K ), we havelim N →∞ f ( x ) + · · · + f ( x N ) N = Z K f d | µ | and lim N →∞ ε f ( x ) + · · · + ε N f ( x N ) N = Z K f dµ. µ -Riemann integrable functions. Now let K be acompact interval of the real line, say for simplicity K = I = [0 , I with (proper) functions ofbounded variation (BV) on I , Theorem 6 yields Theorem 7: Let ϕ : I → R be a BVfunction with ϕ (0) = 0, V ϕ = 1 and ϕ is right continuous on I . Then there aresequences ( x n ) ⊆ I and ( ε n ) ⊆ {± } , such that for every point of continuity x of ϕ wehave lim N →∞ N N X k =1 χ [0 ,x ) ( x k ) = υ ( x ) and lim N →∞ N N X k =1 ε k χ [0 ,x ) ( x k ) = ϕ ( x ) , where υ is the function of total variation of ϕ on I and υ (1) = V ϕ . The last theorempartially generalizes a classical result from [15] (Theorem 8 in our treatment) which saysthat the equalities of Theorem 7 are valid for every x ∈ I , provided that ϕ is increasingwith ϕ (0) = 0 and ϕ (1) = 1 (of course, since ϕ is increasing, we have υ = ϕ and ε k = 1for all k ≥ ϕ on I with ϕ (0) = 0, V ϕ = 1 and for eachpoint x ∈ I . This result is a common generalization of Theorems 7,8 and is the mainresult of the second section. The proof of Theorem 9 is rather elaborate and is presentedin several steps (Lemmas 5,6,7 etc.). We also note that the notion of discrepancy of asequence in I is crucial in its proof. Preliminaries If E is any Banach space, then B E denotes its closed unit ball. A subset L of E is said tobe total in E , if its linear span h L i of L is dense in E . A sequence ( x n ) ⊆ E is said to beCes`aro summable, if the corresponding sequence x + ··· + x n n , n ≥ x n ) converges in norm. Let A ⊆ E , then co ( A ) is the convex hull of A and co ( A ) thenorm closure of co ( A ), which by a classical theorem of Mazur coincides with the weakclosure of co ( A ). If A ⊆ E ∗ , then co w ∗ ( A ) denotes the closure of co ( A ) in the weak ∗ topology of E ∗ . Let x ∈ co ( A ) with x = P nk =1 λ k x k , where x , . . . , x n are distinct pointsof A , λ k > k = 1 , , . . . , n and P nk =1 λ k = 1, then we set suppx = { x , . . . , x n } .Let K be a compact Hausdorff space, then C ( K ) is the Banach space with sup-norm(denoted by k · k ∞ ) of all continuous real valued functions on K . The dual C ( K ) ∗ of C ( K ) is isometrically identified via the classical Riesz representation theorem with thespace M ( K ) of all finite regular signed Borel measures on K , with norm k µ k = | µ | ( K ).By M + ( K ) (resp. P ( K )) we denote the positive (resp. probability) measures on K .When µ ∈ M + ( K ), a bounded function f : X → R is said to be µ -Riemann integrable,if the set of discontinuity points of f has µ -measure zero. It is an easy consequenceof Lusin’s theorem that each µ -Riemann integrable function is µ -measurable and hence µ -integrable.Let X be a (nonempty) set; then | X | denotes the cardinality of X and ℓ ∞ ( X ) theBanach space (with sup-norm) of all bounded real valued functions on X . It is well3nown that ℓ ∞ ( X ) is linearly isometric to the space C ( βX ), where βX is the Stone-ˇCech compactification of the discrete set X . If x ∈ X , then δ x denotes the pointmass at x , that is, the Dirac measure δ x : ℓ ∞ ( X ) → R such that δ x ( f ) = f ( x ), for f ∈ ℓ ∞ ( X ). We denote by F ( X ) the set of probability measures of finite support on X , thus F ( X ) = co ( { δ x : x ∈ X } ); if µ ∈ F ( X ), x , . . . , x n are distinct points of X so that µ ( { x } ) = λ k >
0, for k = 1 , , . . . , n and P nk =1 λ k = 1, then µ = P nk =1 λ k δ x k and thus suppµ = { x , . . . , x n } . We note that if X is compact Hausdorff, then P ( X ) isweak ∗ compact and convex subset of M ( X ) = C ( X ) ∗ and the set of its extreme points exP ( X ) coincides with the set of Dirac measures on X ; therefore by Krein-Milman’stheorem P ( X ) = co w ∗ ( { δ x : x ∈ X } ) = F ( X ) w ∗ .Whilst most of our results remain valid in the complex case, we assume for simplicitythat all Banach spaces (and functions) are real and in certain cases we indicate whathappens in the complex case. We begin by generalizing and improving an important result of Niederreiter, essentiallyfollowing the proof of the original result (see Th.1 of [19]).
Theorem 1.
Let X be a nonempty set, L a subset of the closed unit ball B of ℓ ∞ ( X ) and ( µ j ) ⊆ F ( X ) . Assume that the sequence ( µ j ) converges pointwise on L , that is,there exists a function µ : L → R such that µ j ( f ) −→ j →∞ µ ( f ) ∀ f ∈ L. Then there is a sequence ω = ( x n ) ⊆ S ∞ j =1 suppµ j such that1. The sequence υ N = δ x + ··· + δ xN N −→ N →∞ µ pointwise on L .2. Moreover, if the sequence ( µ j ) converges to µ uniformly on L , then ( υ N ) convergesto µ uniformly on L . The main tool for proving the above theorem is the following Lemma (see Lemma 1of [19]).
Lemma 1.
Let µ ∈ F ( X ) ; then there exists a positive constant C ( µ ) and a sequence ω = ( y n ) in X , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 χ M ( y k ) − µ ( M ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( µ ) N (1) for all N ∈ N and for all subsets M ⊆ X . In particular C ( µ ) = ( m − h m i will do, where m = | suppµ | .
4t is necessary for our purposes to prove that a modification of the above Lemmaholds, not only for characteristic functions, but also for every bounded function f : X → R . We recall that the set of extreme points exB of the unit ball B of ℓ ∞ ( X ) consistsof all functions f : X → R such that | f ( x ) | = 1, for all x ∈ X and the well known factthat B = co ( exB ). Proposition 1.
Let f : X → R be any bounded function. Then inequality (1) of Lemma1 holds in the following modified form(a) If f ∈ co ( exB ) then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f ( y k ) − Z X f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( µ ) N (2) for all N ≥ .(b) If f is any bounded function, then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f ( y k ) − Z X f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k ∞ N (1 + C ( µ )) (3) for all N ≥ .Proof. (a) Assume first that f ∈ exB . Set V = { x ∈ X : f ( x ) = 1 } , then the comple-ment of V is the set V c = { x ∈ X : f ( x ) = − } . Therefore f = χ V − χ V c . So we get for N ∈ N that P Nk =1 f ( y k ) = P Nk =1 χ V ( y k ) − P Nk =1 χ V c ( y k ) and R X f dµ = µ ( V ) − µ ( V c ).Now from Lemma 1 we have, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f ( y k ) − Z X f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 χ V ( y k ) − µ ( V ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 χ V c ( y k ) − µ ( V c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( µ ) N + C ( µ ) N = 2 C ( µ ) N .
It now follows easily from the last inequality that (2) remains valid, for all f ∈ co ( exB )).(b) It is clear that it suffices to prove (3) for f ∈ B . Since B = co ( exB ), there isa sequence ( f n ) ⊆ co ( exB ) such that f n → f uniformly on X . Given N ∈ N , consider n ∈ N such that k f − f n k ∞ < N (4) . Then from assertion (a) and (4) we get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f ( y k ) − Z X f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 ( f ( y k ) − f n ( y k )) + (cid:18)Z X f n dµ − Z X f dµ (cid:19) + N N X k =1 f n ( y k ) − Z X f n dµ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N N X k =1 | f ( y k ) − f n ( y k ) | + Z X | f − f n | dµ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f n ( y k ) − Z X f n dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N · N · N + 1 N µ ( X ) + 2 C ( µ ) N = 2 N (1 + C ( µ )) . Remark 1.
Let f = Ref + iImf be any complex function so that | f ( x ) | = p ( Ref ( x )) + ( Imf ( x )) ≤ for all x ∈ X . Then it is easy to prove that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f ( y k ) − Z X f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ N (1 + C ( µ )) (5) for all N ≥ .In particular (5) is valid for any extreme point f of the unit ball B of the complexBanach space ℓ ∞ ( X ) (recall that the extreme points of B are the functions of the form f : X → C , such that | f ( x ) | = 1 for all x ∈ X ). We now proceed with the proof of Theorem 1. Note that assertion (a) is slightlymore general than Theorem 1 of [19]; assertion (b) is new.
Proof. (of Theorem 1) (1) Assume that ( µ j ) converges pointwise on L to µ . By Lemma1 there exist positive constants C j = C ( µ j ) and sequences ω j = ( x j,n ) n ≥ , j ∈ N suchthat relation (1) of Lemma 1 holds. For each j ∈ N , choose a positive integer r j so that r j ≥ max { j , j ( C + · · · + C j +1 ) } . Put r = 0; we define a sequence ω = ( x n ) as follows.Every positive integer n has a unique representation of the form n = r + r + · · · + r j − + s with j ≥ < s ≤ r j ; we set x n = x j,s . Take an integer N > r ; N can be writtenin the form N = r + · · · + r k + s with 0 < s ≤ r k +1 . For any function f ∈ L we get N X n =1 f ( x n ) = k X j =1 r j X λ =1 f ( x j,λ ) ! + s X λ =1 f ( x k +1 ,λ ) . Therefore | υ N ( f ) − µ ( f ) | = (cid:12)(cid:12)(cid:12) N P Nn =1 f ( x n ) − µ ( f ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X j =1 r j N r j r j X λ =1 f ( x j,λ ) − µ j ( f ) ! + sN s s X λ =1 f ( x k +1 ,λ ) − µ k +1 ( f ) ! + k X j =1 r j N µ j ( f ) + sN µ k +1 ( f ) − µ ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (using Proposition 1) ≤ k X j =1 r j N (cid:20) r j (1 + C j ) (cid:21) + sN · s (1 + C k +1 ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N k X j =1 r j µ j ( f ) + sµ k +1 ( f ) − µ ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ by letting K ( N, f ) = 1 N k X j =1 r j µ j ( f ) + sµ k +1 ( f ) − µ ( f ) r k k +1 X j =1 (1 + C j ) + | K ( N, f ) | = 2 r k ( k + 1) + 2 r k k +1 X j =1 C j + | K ( N, f ) | ≤ (since r k ≥ max { k , k ( C + · · · + C k +1 ) } ) ≤ k + 1) k + 2 k + | K ( N, f ) | . If N → ∞ then k → ∞ and the sum of the first two terms tends to zero. In order toprove that the third term tends to zero, we set for every N > r A N = (cid:16) r N , r N , · · · , r k N , sN , , · · · (cid:17) , where N = r + r + · · · + r k + s, < s ≤ r k +1 . Then A = ( A N ) defines an infinitereal matrix that is a regular method of summability. If we set H f = ( µ j ( f )) j ≥ , where f ∈ L , then we have A N · H f = 1 N k X j =1 r j µ j ( f ) + sµ k +1 ( f ) , N ≥ . Since µ j ( f ) −→ j →∞ µ ( f ) for f ∈ L and A is a regular method of summability, we get that | A N · H f − µ ( f ) | = | K ( N, f ) | −→ N →∞ ∀ f ∈ L . So we are done.(2) We assume now that ( µ j ) converges to µ uniformly on L . Since A is a regularmethod of summability, we get that A N · H f −→ N →∞ µ ( f ) uniformly on L. Therefore given ε >
0, there is N = N ( ε ) such that N ≥ N ⇒ | A n · H f − µ ( f ) | = | K ( N, f ) | ≤ ε ∀ f ∈ L and of course 2 (cid:0) k +1 k + k (cid:1) ≤ ε , if N is quite large. It then follows from the above that N ≥ N ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X n =1 f ( x n ) − µ ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε ∀ f ∈ L, which means that υ N = δ x + · · · + δx N N −→ N →∞ µ uniformly on L . 7 emark 2. We notice that using inequality (5) of Remark 1 in the proof of Th. 1instead of inequality (3) of Prop.1, we can easily prove that Th.1 is also valid assumingthat L consists of complex functions. Therefore Th.2, which we are going to prove, andeverything depending on this theorem is also valid in the complex case. The rest of this section is devoted to some applications of the previous results (Prop.1and Th.1) in Banach space theory. We first prove the following
Theorem 2.
Let E be a Banach space, F a bounded subset of E so that F ⊆ B (0 , R ) and x ∈ E . Then we have:1. Assume that x ∈ co ( F ) and let F ⊆ F be any finite set so that x ∈ co ( F ) . Thenthere is a sequence ( x n ) ⊆ F and a positive constant C = C ( | F | ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N X k =1 x k − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ RN (1 + C ) for all N ≥ in particular lim N →∞ N N X k =1 x k k·k = x.
2. Assume that x ∈ co ( F ) . Then there is a sequence ( x n ) ⊆ F such that lim N →∞ N N X k =1 x k k·k = x. Proof.
Assume without loss of generality that R = 1, that is F ⊆ B E ; otherwise wereplace F by R F and x by R x . We set X = B E and notice that each f ∈ E ∗ can beidentified with a bounded (continuous) function on X through the isometry operator T : f ∈ E ∗ T ( f ) = f /X ∈ ℓ ∞ ( X ).(1) Let x ∈ co ( F ), then x = α y + · · · + α m y m , where y , . . . , y m are distinct pointsof F , α k > k = 1 , , . . . , m and P mk =1 α k = 1. We can consider x as a finitelysupported measure µ on X , by letting µ = P mk =1 α k δy k ; clearly suppµ = { y , . . . , y m } .Then µ represents x , that is, for every f ∈ B E ∗ Z X f dµ = m X k =1 α k f ( y k ) = f m X k =1 α k y k ! = f ( x ) . It then follows from Prop.1 (see also Remark 1) that there is ( x n ) ⊆ F , where F = supp ( µ ), such that for every f ∈ ℓ ∞ ( X ) with k f k ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f ( x k ) − Z X f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (1 + C ( µ )) for all N ≥ .
8n particular, if f ∈ E ∗ with k f k ≤
1, then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 f ( x k ) − Z X f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f N N X k =1 x k − x !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (1 + C ( µ )) for all N ≥ , which implies that for all N ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N X k =1 x k − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f N N X k =1 x k − x !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : f ∈ B E ∗ ) ≤ N (1 + C ( µ )) . We set C = C ( µ ), hence C depends on m = | F | and obtain the desired result.(2) Let x ∈ co ( F ); then there is a sequence ( µ j ) of convex combinations of elementsof F such that µ j k·k −→ x. (6)We consider each µ j as a finitely supported probability measure on F ⊆ X = B E .So if we set L = T ( B E ∗ ) ⊆ B ℓ ∞ ( X ) , then (6) means that µ j −→ j →∞ x uniformly on L . Itthen follows from Theorem 1(2) that there is a sequence ( x n ) ⊆ ∪ ∞ j =1 suppµ j ⊆ F suchthat the sequence of arithmetic means δ x + · · · + δ x N N −→ N →∞ x uniformly on L, equivalently x + ··· + x N N k·k −→ N →∞ x . The proof of the theorem is complete.Assertion (1) of the above theorem has a strong relationship with a Lemma due toMaurey (see Lemma D of [5]) which states that Lemma 2.
Let E be a Banach space of type p for some p > , F ⊆ E and x ∈ co ( F ) .Set q = pp − . Then for every N ∈ N there exist x , . . . , x N ∈ F such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N X k =1 x k − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ diam ( F ) T p ( E ) N q . ( T p ( E ) is the type p constant of E , see pp. 137-8 of [1]). In our case the constant C depends on the cardinality of the finite subset of F thatsupports x ; in Maurey’s Lemma it depends only on the space E , which must be of type p . If we assume, as we may, that F ⊆ B E (thus diam ( F ) ≤
2) and N is quite large,then clearly CN ≤ T p ( E ) N q , hence we get Maurey’s Lemma. In some way assertion (1) ofTh.2 is the ”pointwise” version of Maurey’s Lemma.From assertion (2) of Th.2 together with Mazur’s classical result, stating that theweak and the norm closure of any convex subset of a Banach space coincide, we obtainthe following interesting consequence. 9 roposition 2. Let E be a Banach space and ( y n ) ⊆ E be any weakly convergentsequence, so that y n w → y . Then there is a function ϕ : N → N such that the sequence x n = y ϕ ( n ) , n ≥ is Ces`aro summable to y , that is x + ··· + x N N k·k −→ N →∞ y .Proof. The set F = { y n : n ≥ } is bounded. Since y ∈ F w , we get that y ∈ co w ( F ).By Mazur’s Theorem we have that co w ( F ) = co ( F ). So y ∈ co ( F ) and then assertion(2) of Th.2 can be applied. Remark 3.
It is well known that the function ϕ of Prop.2 cannot in general be chosenstrictly increasing (neither 1-1). In fact, it is possible to find a weakly null sequence ( y n ) , such that for every subsequence ( y ′ n ) of ( y n ) the sequence of arithmetic means y ′ + ··· + y ′ N N , N ≥ is not norm convergent. The first such example was constructed by J.Schreier (see [17] and [3]). We note in this connection that in [3] is given a completeclassification of the complexity of weakly null sequences, by the aid of a hierarchy ofsummability methods introduced there. Still another immediate but useful consequence of assertion (2) of Th.2 is the follow-ing
Proposition 3.
Let E be a Banach space and Ω be a closed convex bounded subsetof E , so that Ω is equal to the closed convex hull of its extreme points ex Ω , that is Ω = co ( ex Ω) . Then for every x ∈ Ω , there is a sequence ( x n ) ⊆ ex Ω which is Ces`arosummable to x . We now present some applications of Prop.3.
Corollary 1.
Let Ω be a weakly compact and convex subset of a Banach space E (inparticular Ω = B E and E is reflexive). Then for every x ∈ Ω there is a sequence ( x n ) of extreme points of Ω which is Ces`aro summable to x .Proof. Since Ω is weakly compact and convex, by Krein-Milman’s Theorem we havethat Ω = co ( ex Ω). Hence the result is an immediate consequence of Prop.3.The next result concerns Banach spaces of the form C ( K ), where K is a compactHausdorff space. We recall that K is called totally disconnected, if it has a base for itstopology consisting of open and closed (clopen) sets. Corollary 2.
Let K be a compact Hausdorff space. We assume that either(a) C ( K ) is the space of continuous complex functions on K , or(b) K is totally disconnected and C ( K ) is the space of continuous real functions on K .Then for every f ∈ B = B C ( K ) , there exists a sequence ( f n ) of extreme points of B such that f + · · · + f N N k·k ∞ −→ N →∞ f. roof. In either case we have that B = co ( exB ) (see Ths. 1.6 and 1.8 of [4]). HenceProp.3 can be applied. Remark 4. (a) Recall that if K is a compact Hausdorff space, then f ∈ exB iff | f ( x ) | =1 , for all x ∈ K (see Th. 1.3 of [4]). If K is in addition totally disconnected, C ( K ) isthe space of real continuous functions on K and f ∈ exB , then the sets V = { x ∈ K : f ( x ) = 1 } and V c = { x ∈ K : f ( x ) = − } constitute a partition of K in two clopensets. Thus the extreme points of B are completely determined by the clopen nonemptysubsets of K .(b) We have already used the above remark in the special case of the Banach space ℓ ∞ ( X ) (cf. the proof of Prop.1). Indeed ℓ ∞ ( X ) is isometric to C ( βX ) , where βX is the Stone-ˇCech compactification of the discrete set X , which is a compact extremally disconnectedspace. We continue our investigation, applying Th.1 to the weak ∗ topology of the dual E ∗ ofa separable Banach space E . As we shall see, results similar to Th.2 (2) and Prop.3 arevalid. Moreover, our approach has interesting applications for the dual M ( K ) = C ( K ) ∗ of C ( K ), where K is any compact metric space. Proposition 4.
Let E be a separable Banach space, F a bounded subset of its dual E ∗ and f ∈ co w ∗ ( F ) . Then there is a sequence ( f n ) ⊆ F such that f + · · · + f N N w ∗ −→ N →∞ f. Proof.
The proof is similar to the proof of Th.2 (2). We set X = B E ∗ and assumewithout loss of generality that F ⊆ X . Note that, since E is separable, X is weak ∗ compact and metrizable and also that each x ∈ E can be identified with a continuousfunction on X through the linear isometry T : x ∈ E T ( x ) = x/X ∈ C ( X ) ⊆ ℓ ∞ ( X ) . Since the weak ∗ closed convex hull co w ∗ ( F ) ⊆ X is a weak ∗ compact and metrizable set,given any f ∈ co w ∗ ( F ) there is a sequence ( µ j ) of convex combinations of elements of F such that µ j w ∗ −→ f (7) . We cosider each µ j as a finitely supported probability measure on F ⊆ X . So if we set L = T ( B E ) ⊆ B ℓ ∞ ( X ) then (7) means that µ j → f pointwise on L .It then follows from Th.1 (1) that there is a sequence ( f n ) ⊆ ∪ ∞ j =1 suppµ j ⊆ F suchthat δ f + · · · + δ f N N −→ N →∞ f pointwise on L ;equivalently f + ··· + f N N w ∗ −→ N →∞ f . 11 roposition 5. Let E be a separable Banach space and Ω be a weak ∗ compact andconvex subset of E ∗ . Then for every f ∈ Ω there is a sequence ( f n ) ⊆ ex Ω such that f + · · · + f N N w ∗ −→ N →∞ f. Proof.
It follows immediately from Krein-Milman’s theorem and Prop.4.Since the dual unit ball B E ∗ of any Banach space E is a weak ∗ compact and convexset, we immediately obtain the following Corollary 3.
Let E be a separable Banach space. Then for every f ∈ B E ∗ , there is asequence ( f n ) of extreme points of B E ∗ such that f + · · · + f N N w ∗ −→ N →∞ f. An immediate consequence of Prop.5 is the following well known result.
Theorem 3.
Let K be any compact metric space. Then every Borel probability measure µ on K (i.e. µ ∈ P ( K ) ) admits a u.d. sequence.Proof. Since K is compact Hausdorff, we have that P ( K ) is a weak ∗ compact and convexsubset of C ( K ) ∗ = M ( K ) with exP ( K ) = { δ x : x ∈ K } , thus P ( K ) = co w ∗ ( { δ x : x ∈ K } ). K is a metrizable space, hence C ( K ) is separable and so Prop.5 can be applied.Applying corollary 3 to C ( K ) ∗ , where K is a compact metric space, gives the fol-lowing result that seems to be new and will be our motivation for the next section. Theorem 4.
Let K be any compact metric space. Then for every µ ∈ B = B C ( K ) ∗ there are sequences ( x n ) ⊆ K and ( ε n ) ⊆ {± } such that ε δ x + · · · + ε N δ x N N w ∗ −→ N →∞ µ. (In case when C ( K ) is the space of continuous complex functions, ( ε n ) ⊆ T = { z ∈ C : | z | = 1 } ) . Proof.
We first assume that C ( K ) is the space of continuous real functions. We thenhave exB = {± δ x : x ∈ K } and hence B = co w ∗ ( {± δ x : x ∈ K } ).In the complex case we have that exB = { αδ x : x ∈ K and α ∈ C , | α | = 1 } and hence B = co w ∗ ( { αδ x : x ∈ K and α ∈ C , | α | = 1 }} ) (see Th.1.9 of [4]). So the result followsimmediately from Corollary 3. Remark 5. (a) Theorem 3 is of course a direct consequence of Niederreiter’s mainresult (see Th.2 of [19]). We state Theorem 3 here, because it shows that Prop.5 canbe considered as a generalization of such a well known result to the much wider class ofseparable Banach spaces.(b) A compact Hausdorff space K is said to be angelic iff for every A ⊆ K andeach x ∈ A there is a sequence ( x n ) ⊆ A such that x n → x . It is clear that, if the ual unit ball ( B E ∗ , w ∗ ) of a Banach space E is an angelic space, then Prop.4 andits consequences (Prop.5, Cor.3 and Ths 3,4) remain valid. Well known classes of(not necessarily separable) Banach spaces with angelic dual balls are weakly compactlygenerated (WCG) and their generalizations, like weakly countably determined (WCD)Banach spaces, etc. (see [2], [10] and [14]).We also note that if K is any compact Hausdorff space so that the convex hull co ( { δ x : x ∈ K } ) is weak ∗ sequentially dense in P ( K ) , then it is easy to see that every µ ∈ M ( K ) with k µ k = 1 satisfies the conclusion of Theorem 4 (in particular, by a resultof Niederreiter mentioned in (a), every µ ∈ P ( K ) admits a u.d. sequence).Finally, asuming Martin’s axiom plus the negation of Continuum Hypothesis (MA+ ¬ CH), Theorem 4 remains valid for every compact separable space K of topological weight w ( K ) < c , where c is the cardinality of the continuum (the proof is essentially the sameas the proof of Prop. 2.21 of [18]).(c) Let E be a Banach space not containing an isomorphic copy of ℓ . Then by aresult of Haydon, every weak ∗ compact and convex subset Ω of E ∗ is the norm closedconvex hull of its extreme points (see [13]). It then follows from this result and theaforementioned considerations that for every f ∈ Ω there exists a sequence ( f n ) ⊆ ex Ω norm Ces`aro summable to f . It follows in particular that if E is of the form C ( K ) , where K is compact and Hausdorff (since ℓ * C ( K ) , every µ ∈ P ( K ) is purely atomic), thenfor every µ ∈ P ( K ) there is a µ -u.d. sequence ( x n ) in K with the stronger property δ x + · · · + δ x N N k·k ∞ −→ N →∞ µ. Note that the last result can also be proved by a direct method. We also note that a classof (not necessarily separable) Banach spaces not containing ℓ is that of Asplund spaces;a Banach space E is called Asplund, if every separable subspace of E has separable dual(see [10]). In this section we shall concentrate on duals C ( K ) ∗ = M ( K ) of Banach spaces of theform C ( K ) with K compact (metrizable) space and shall further investigate the effectof the results of the previous section to the uniform distribution of sequences in K .Theorem 4 inspires the following generalization of the classical concept of uniformlydistributed sequences defined for regular Borel probability measures on compact spaces(see Def.1.1 of [15]). Definition 1.
Let K be a compact Hausdorff space and µ ∈ M ( K ) with total variation | µ | ( K ) = 1 (i.e. µ is a regular Borel signed measure with k µ k = 1). We say that asequence ( x n ) ⊆ K is µ -u.d. if both of the following conditions are satisfied1. ( x n ) is | µ | -u.d. (in the classical sense) and2. there is a sequence of signs ( ε n ) ⊆ {± } such that ε δ x + · · · + ε N δ x N N w ∗ −→ N →∞ µ N →∞ ε f ( x )+ ··· + ε N f ( x N ) N = R K f dµ , for all f ∈ C ( K )).We note that:(a) By applying the above equality for the constant function f = 1, we get that µ ( K ) = R K dµ = lim N →∞ ε + ··· + ε N N and(b) if µ ∈ P ( K ) admits a u.d. sequence ( x n ), then we can take ε n = 1 for all n ≥ µ ∈ M ( K ) with k µ k = 1 (= | µ | ( K )). It then follows from Radon-Nikodymtheorem that there is a Borel function h : K → R with | h ( x ) | = 1, for all x ∈ K suchthat dµ = hd | µ | ⇔ h = dµd | µ | . The function h is the so-called Radon-Nikodym derivative of µ with respect to its totalvariation | µ | . So we have that Z K f dµ = Z K f hd | µ | , for all bounded Borel measurable functions f : K → R .With the above notation we have the following Proposition 6.
Assume that | µ | admits a u.d. sequence ( x n ) (in the classical sense)and also that the function h is | µ | -Riemann integrable. Then the sequence ( x n ) is µ -u.d.(in the sense of Def.1).Proof. Let f ∈ C ( K ); then the function f h is | µ | -Riemann integrable and since ( x n ) is | µ | -u.d., we get that Z K f dµ = Z K f hd | µ | = lim N →∞ ( f h )( x ) + · · · + ( f h )( x N ) N .
So the desired sequence of signs is the sequence ε n = h ( x n ) , n ≥ Remark 6.
Later in this section, we shall present a class of measures µ ∈ P ( K ) ,where K = [ a, b ] is a compact interval of the real line, so that the function h = dµd | µ | is | µ | -Riemann integrable. It follows in particular from Prop.6 and standard results (cf. Th.2.2 of [15]) that,whenever K is compact metric, µ ∈ M ( K ) with k µ k = 1 so that the function h = dµd | µ | is | µ | -Riemann integrable, then µ admits a u.d. sequence in the sense of Def.1. In thesequel we are going to show, essentially by the method of proof of Th.2.2 of [15], thatevery measure µ ∈ M ( K ) with k µ k = 1 admits a u.d. sequence.We first cite some preliminaries. Let K be a compact Hausdorff space; we denoteby K ∞ the cartesian product of countably many copies of K . Then K ∞ is a compactHausdorff space endowed with the product topology. If µ ∈ P ( K ), then µ induces theproduct measure µ ∞ in K ∞ , which we may assume to be complete. We also denoteby B ( K ) the Banach space of bounded Borel functions on K endowed with supremumnorm. 14 heorem 5. Let K be a compact metric space and µ ∈ M ( K ) with k µ k = 1 ; also let S be the set of all sequences in K , which are µ -u.d. considered as a subset of K ∞ . Then µ ∞ ( S ) = 1 .Proof. We consider a countable total subset L = { f n : n ≥ } of C ( K ) with f ≡ h = dµd | µ | (=the Radon-Nikodym derivative of µ with respect to | µ | ). Set M = L ∪ hL = { f n : n ≥ } ∪ { f n h : n ≥ } . As the members of the set M are bounded Borelfunctions and | µ | ∈ P ( K ), for each g ∈ M there is a | µ | ∞ -measurable subset B g of K ∞ with | µ | ∞ ( B g ) = 1 such thatlim N →∞ N N X k =1 g ( x k ) = Z K gd | µ | ∀ ( x , . . . , x k , . . . ) ∈ B g (8)(see Lemma 2.1, p.182 of [15]).Set B = ∩ g ∈ M B g ; as the set M is countable, we get that | µ | ∞ ( B ) = 1. Let( x , . . . , x k , . . . ) ∈ B . Since the set L is total in C ( K ), we get that this sequenceis | µ | -u.d. in K , that is (8) is valid for every f ∈ C ( K ). Note that the operator T : f ∈ C ( K ) hf ∈ B ( K ) is a linear isometry, thus the set hL is total in the closedsubspace T ( C ( K )) of the Banach space B ( K ). So we get that equation (8) also holdsfor each member of the space T ( C ( K )), that islim N →∞ N N X k =1 ε k f ( x k ) = Z K f hd | µ | = Z K f dµ ∀ f ∈ C ( K ) (9)where ε k = h ( x k ) , k = 1 , , . . . . So we are done. Remark 7.
Note that equalities (1) and (2) of Definition 1 (for f ∈ C ( K ) ) are equiv-alent to the following lim N →∞ (1 + ε ) f ( x ) + · · · + (1 + ε N ) f ( x N )2 N = Z K f dµ + (10) and lim N →∞ (1 − ε ) f ( x ) + · · · + (1 − ε N ) f ( x N )2 N = Z K f dµ − (11) , where µ + = ( | µ | + µ ) and µ − = ( | µ | − µ ) are the positive and negative variations ofthe measure µ .Indeed, assuming that (1) and (2) of Def.1 are valid, for f ∈ C ( K ) we have Z K f dµ + = 12 (cid:20)Z K f d ( | µ | + µ ) (cid:21) = 12 (cid:20)Z K f d | µ | + Z K f dµ (cid:21) == lim N →∞ (1 + ε ) f ( x ) + · · · + (1 + ε N ) f ( x N )2 N so (10) holds. In a similar way we get equality (11).In the converse direction, by adding and subtracting (10) and (11) we get (1) and(2) of Def.1 respectively. µ -Riemann integrable function (i.e. abounded function f : K → R that is | µ | -Riemann integrable). In order to prove this, weshall use the following well known facts:Fact I Let ν ∈ M + ( K ) (with ν ( K ) > f : K → R is ν -Riemann integrable iff for every ε > f , f ∈ C ( K ) such that f ≤ f ≤ f and Z K ( f − f ) dν ≤ ε. (see p.90 of [18])Fact II Let ν ∈ P ( K ) and ( x n ) ⊆ K be a ν -u.d. sequence. Then for every ν -Riemannintegrable function f : K → R lim N →∞ N N X k =1 f ( x k ) = Z K f dν. The proofs of Facts I and II are essentially contained in the proofs of Th.1, ch.1 andTh.2, ch.3 (see also ex.1.12, p.179) of [15].Let us prove, for instance, equality (10). So let f : K → R be any µ -Riemannintegrable function and ε >
0. Then by Fact I there are f ≤ f ≤ f continuousfunctions, such that 0 ≤ Z K ( f − f ) d | µ | ≤ ε. Since 0 ≤ ± h ≤ (cid:16) h = dµd | µ | (cid:17) we get that F = f (cid:18) h (cid:19) ≤ F = f (cid:18) h (cid:19) ≤ F = f (cid:18) h (cid:19) and0 ≤ Z K ( F − F ) d | µ | = Z K ( f − f ) (cid:18) h (cid:19) d | µ | ≤ Z K ( f − f ) d | µ | ≤ ε. Set I = R K F d | µ | (cid:0) = R K f (cid:0) h (cid:1) d | µ | = R K f dµ + (cid:1) . We then have that I − ε = Z K F d | µ | − ε ≤ Z K F d | µ | = Z K f dµ + == lim N →∞ N N X k =1 (1 + ε k ) f ( x k ) ≤ lim inf N →∞ N N X k =1 (1 + ε k ) f ( x k ) ≤≤ lim sup N →∞ N N X k =1 (1 + ε k ) f ( x k ) ≤ lim N →∞ N N X k =1 (1 + ε k ) f ( x k ) == Z K f dµ + = Z K F d | µ | ≤ Z K F d | µ | + ε = I + ε. Since ε is arbitrarily small, we get (10).Taking into account the above remarks and Th.5, we obtain the following result16 heorem 6. Let K be a compact metric space and µ ∈ M ( K ) with k µ k = 1 . Thenthere is a sequence ( x n ) ⊆ K and a sequence of signs ( ε n ) ⊆ {± } (where ε n = h ( x n ) and h = dµd | µ | ) such that, for every µ -Riemann integrable function f : K → R we have ( a ) lim N →∞ N P Nk =1 f ( x k ) = R K f d | µ | ; in particular ( x n ) is | µ | -u.d. ( b ) lim N →∞ N P Nk =1 ε k f ( x k ) = R K f dµ, ( c ) lim N →∞ N P Nk =1 (1 + ε k ) f ( x k ) = R K f dµ + and ( d ) lim N →∞ N P Nk =1 (1 − ε k ) f ( x k ) = R K f dµ − . In the sequel, we focus on the special case of Theorem 6 when K is a compact intervalof the real line, say K = [ a, b ]. Let ϕ : [ a, b ] → R be a function of bounded variation. Wedenote by υ, p and n the (increasing) functions of total, positive and negative variationof ϕ ( υ ( x ) = V xa ϕ, x ∈ [ a, b ]).We note that these functions are connected as follows: p ( x ) = 12 ( υ ( x ) + ϕ ( x ) − ϕ ( a )) and n ( x ) = 12 ( υ ( x ) − ϕ ( x ) + ϕ ( a )) , (see p.208 of [7]).We recall that the space M ( K ) of signed Borel measures on K is in one-to-onecorrespondence with the space of functions of bounded variation on K which are rightcontinuous on ( a, b ) with ϕ ( a ) = 0, in the sense that each µ ∈ M ( K ) is uniquely definedby such a ϕ by the rule µ (( y, x ]) = ϕ ( x ) − ϕ ( y ) , for a ≤ y < x ≤ b (see Th. 3.29 of [9] and Th. 14.26 of [7]).With the above notation and terminology, Theorem 6 yields the following Theorem 7.
Let ϕ : [ a, b ] → R be a right continuous function (of bounded variation)with total variation υ ( b ) = V ba ϕ = 1 and ϕ ( a ) = 0 . Then there are sequences ( x n ) ⊆ [ a, b ] and ( ε n ) ⊆ {± } , such that for every point of continuity x ∈ [ a, b ] of ϕ we have ( a ) lim N →∞ N P Nk =1 χ [ a,x ) ( x k ) = υ ( x )( b ) lim N →∞ N P Nk =1 ε k χ [ a,x ) ( x k ) = ϕ ( x ) , ( c ) lim N →∞ N P Nk =1 (1 + ε k ) χ [ a,x ) ( x k ) = p ( x ) and ( d ) lim N →∞ N P Nk =1 (1 − ε k ) χ [ a,x ) ( x k ) = n ( x ) . roof. Let µ = µ ϕ be the signed (Lebesgue-Stieljes) measure defined by ϕ on [ a, b ] bythe rule µ (( a, x ]) = ϕ ( x ) (= µ ([ a, x ])) for x ∈ ( a, b ]. As is well known, the Jordandecomposition and the total variation of µ are given by µ = µ + − µ − , | µ | = µ + + µ − where | µ | = µ υ , µ + = µ p and µ − = µ n , thus in particular k µ k = υ ( b ) = 1 (see Th. 3.29and exs. 28, 29, p.107 of [9]).Let D ϕ be the (countable) set of discontinuity points of ϕ ; then every interval I ⊆ [ a, b ] whose both endpoints do not belong to D ϕ has characteristic function which is µ -Riemann integrable. Then by applying Theorem 6 to intervals of the form [ a, x ), with x / ∈ D ϕ we get the conclusion (of course ε n = h ( x n ) , n ≥ h = dµd | µ | ).The last theorem partially generalizes an important result from [15] (Th. 4.3, p.138)stating that: Theorem 8.
Let ϕ : I = [0 , → R be an increasing function, with ϕ (0) = 0 and ϕ (1) = 1 . Then there is a sequence ( x n ) ⊆ I , such that lim N →∞ N N X k =1 χ [ a,x ) ( x k ) = ϕ ( x ) , for ≤ x ≤ . We then say that ( x n ) has ϕ as the asymptotic distribution function mod 1 (abbreviateda.d.f.(mod 1) ϕ ( x ) ). The proof of this result is given in two steps. First, the continuous case is proved(Lemma 4.2, p.137 of [15]) and then the general case follows, using a result from realanalysis (Lemma 4.3, p.138 of [15]). We state both of these results for the reader’sconvenience.
Lemma 3.
Let ϕ : I → R be a continuous increasing function, with ϕ (0) = 0 and ϕ (1) = 1 . Then there is a sequence ( x n ) ⊆ I , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 χ [0 ,x ) ( x k ) − ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ log ( N + 1) N log for all N ≥ and ≤ x ≤ . Before we state the real analysis Lemma, we recall that a continuous function ϕ :[ a, b ] → R , ( a, b ∈ R , a < b ) is said to be polygonal (or piecewise linear), if its graphconsists of finitely many straight line segments. Lemma 4.
Let ϕ : [ a, b ] → R be an increasing function. Then there is a sequence ( ϕ k ) of polygonal increasing functions defined on [ a, b ] , satisfying ϕ k ( a ) = ϕ ( a ) and ϕ k ( b ) = ϕ ( b ) for k ≥ , which converges pointwise to ϕ , that is, lim k →∞ ϕ k ( x ) = ϕ ( x ) ,for all x ∈ [ a, b ] . ϕ : [ a, b ] → R of boundedvariation (with ϕ ( a ) = 0 and V ba ϕ = 1) and for each point x ∈ [ a, b ]. We start bygeneralizing Lemma 4. Lemma 5.
Let ϕ : [ a, b ] → R be a function of bounded variation (with V ba ϕ > ). Also,let υ, p and n be the functions of total, positive and negative variation of ϕ . Then thereare sequences of increasing polygonal functions ( g n ) and ( h n ) defined on [ a, b ] , such thatif we let ϕ k = g k − h k , for k ≥ , then we have that ( ϕ k is polygonal and)(a) g k → p, h k → n and (thus) ϕ k → ϕ pointwise on [ a, b ] ; moreover g k ( a ) = p ( a ) , g k ( b ) = p ( b ) and n k ( a ) = n ( a ) , n k ( b ) = n ( b ) , for k ≥ .(b) If υ k denotes the function of total variation of ϕ k , then υ k is polygonal and υ k → υ pointwise on [ a, b ] .Proof. For each k ≥ P k = { t k = a < t k < · · · < t km k = b } of [ a, b ] with t ki +1 − t ki < k , for 0 ≤ i < m k , that contains all points x ∈ ( a, b ) with υ ( x + 0) − υ ( x − > k (since υ is increasing, there can only be finitely many such x ).We define the functions g k and h k as follows: Set g k ( t ki ) = p ( t ki ) and h k ( t ki ) = n ( t ki ),for 0 ≤ i ≤ m k and then extend g k and h k on [ a, b ] so as to be linear on the intervals[ t ki , t ki +1 ] , ≤ i < m k . Then clearly g k and h k are polygonal and increasing on [ a, b ] and(hence) ϕ k is polygonal on [ a, b ].(a) We shall prove that ( g k ) converges pointwise to p (the proof for ( h k ) is analogous).This is trivial for the endpoints a and b . Let x ∈ ( a, b ); assume first that x is adiscontinuity point of υ . Then υ ( x + 0) − υ ( x − > k on, we willhave that x = t ki k with 0 < i k < m k . Therefore g k ( x ) = p ( x ) for sufficiently large k .Now let υ be continuous at x , then ϕ, p, n are also continuous at x . So let ε > k (say k ≥ k ) we will have y ∈ (cid:18) x − k , x + 1 k (cid:19) ⇒ p ( y ) ∈ ( p ( x ) − ε, p ( x ) + ε ) . Yet for each k we have t ki ≤ x ≤ t ki +1 , for some i = i ( k ) with 0 ≤ i < m k . Since0 < t ki +1 − t ki < k , both t ki , t ki +1 lie in (cid:0) x − k , x + k (cid:1) . Hence, for k ≥ k we obtain g k ( t ki ) = p ( t ki ) > p ( x ) − ε and g k ( t ki +1 ) = p ( t ki +1 ) < p ( x ) + ε ;since g k is increasing, we get that g k ( t ki ) ≤ g k ( x ) ≤ g k ( t ki +1 )and so p ( x ) − ε < g k ( x ) < p ( x ) + ε for k ≥ k , which shows that g k ( x ) → p ( x ).(b) Let x ∈ ( a, b ] (clearly υ ( a ) = υ k ( a ) = 0 for k ≥ ϕ k is a polygonal andhence piecewise C function, we get that υ k ( x ) = Z xa | ϕ ′ k ( t ) | dt = i ( k ) − X λ =0 | ϕ k ( t kλ +1 ) − ϕ k ( t kλ ) | + | ϕ k ( t ki ( k ) ) − ϕ k ( x ) | =19where x ∈ ( t ki ( k ) , t ki ( k )+1 ] and 0 ≤ i ( k ) < m k )= i ( k ) − X λ =0 | ϕ ( t kλ +1 ) − ϕ ( t kλ ) | + | ϕ ( t ki ( k ) ) − ϕ k ( x ) | ≤ V t i ( k ) a ϕ + | ϕ ( t ki ( k ) ) − ϕ k ( x ) | (12) . It is clear that V t i ( k ) a ϕ + | ϕ ( t ki ( k ) ) − ϕ ( x ) | ≤ V xa ϕ ; (13)since ϕ k ( x ) → ϕ ( x ), we get from (12) and (13) thatlim sup k →∞ υ k ( x ) ≤ lim sup k →∞ ( V t i ( k ) a ϕ + | ϕ ( t ki ( k ) ) − ϕ k ( x ) | ) == lim sup k →∞ ( V t i ( k ) a ϕ + | ϕ ( t ki ( k ) ) − ϕ ( x ) | ) ≤ V xa ϕ. (14)But since ϕ k → ϕ pointwise on [ a, b ], we have that υ ( x ) = V xa ϕ ≤ lim inf k →∞ υ k ( x ) (15)(see ex.12, p.205 of [7]).It then follows from (14) and (15) that υ ( x ) = lim k →∞ υ k ( x ) , for x ∈ ( a, b ] . We finally note that it is easy to verify that the function of total variation of a polygonalfunction is also polygonal. Therefore each υ k is a polygonal (and increasing) function.We note that the proof of claim (a) of Lemma 5 is similar to the proof of Lemma 4. Remark 8. (1) Regarding the previous Lemma, we set p k = 12 ( υ k + ϕ k − ϕ k ( a )) and n k = 12 ( υ k − ϕ k + ϕ k ( a )) , where υ k is the function of total variation of ϕ k . Then we have that:(a) p k and n k are the positive and negative variations of ϕ k ,(b) the function ϕ k is polygonal, the functions υ k , p k , n k are polygonal and increasingand(c) ϕ k → ϕ, υ k → υ pointwise on [ a, b ] and hence p k → p, n k → n pointwise on [ a, b ] .(2) Assume now that ϕ ( a ) = 0 and V ba ϕ = 1 . We then have that ϕ k ( a ) = 0 for k ≥ and V ba ϕ k = υ k ( b ) −→ k →∞ υ ( b ) = V ba ϕ = 1 . Now we define Φ k = ϕ k υ k ( b ) , Υ k = υ k υ k ( b ) , P k = p k υ k ( b ) , N k = n k υ k ( b ) and notice the following:(a) Υ k , P k , N k are the functions of total, positive and negative variation of Φ k , so that k ( a ) = 0 and Υ k ( b ) = V ba Φ k = 1 , for k ≥ .(b) Φ k is polygonal and Υ k , P k , N k are polygonal and increasing.(c) For every x ∈ [ a, b ] we have that Φ k ( x ) → ϕ ( x ) , Υ k ( x ) → υ ( x ) and hence P k ( x ) → p ( x ) and N k ( x ) → n ( x ) . It follows from the aforementioned remark that Lemma 5 can be stated as follows:
Proposition 7.
Let ϕ : [ a, b ] → R be a function of bounded variation with ϕ ( a ) = 0 and V ba ϕ = 1 . Also, let υ, p and n be the functions of total, positive and negative variationof ϕ . Then there is a sequence ϕ k : [ a, b ] → R , k ≥ of polygonal functions, such that if υ k , p k and n k are the total, positive and negative variations of ϕ k , then (these functionsare polygonal and)(a) ϕ k → ϕ , υ k → υ , p k → p and n k → n pointwise on [ a, b ] .(b) ϕ k ( a ) = 0 and V ba ϕ k = 1 , for k ≥ . Let ϕ : I = [0 , → R be a continuous function of bounded variation with ϕ (0) = 0and V ϕ = 1. Denote, as usual, the function of total variation of ϕ by υ . Then byTheorem 7, there are sequences ω = ( x n ) ⊆ I and ε = ( ε n ) ⊆ {± } such that( a ) lim N →∞ N P Nk =1 χ [0 ,x ) ( x k ) = υ ( x ) and( b ) lim N →∞ N P Nk =1 ε k χ [0 ,x ) ( x k ) = ϕ ( x ) , for 0 ≤ x ≤ discrepancy D N ( ω ; υ ) of ω = ( x n ) with respect to the (con-tinuous) function υ by the rule D N ( ω ; υ ) = sup ≤ a
We first define the discrepancies of the functions p and n in the obvious way. Itis easy to prove that D N ( ω ; ϕ ) ≤ D N ( ω ; p ) + D N ( ω, n ) (16) . p and n are continuous and hence uniformly continuous on the compact interval I , given any ε > m ∈ N such that x, y ∈ I and | x − y | < m ⇒ | p ( x ) − p ( y ) | < ε | n ( x ) − n ( y ) | < ε . (17)We may pick m quite large, so that m < ε .For such an integer m , set I k = (cid:2) km , k +1 m (cid:1) , ≤ k ≤ m −
1. Using the equalities (c)and (d) of Th.7, we can get a N = N ( m ) ∈ N , such that for every N ≥ N and each k = 0 , , . . . , m we have µ + ( I k ) − m ≤ N N X λ =1 (cid:18) ε λ (cid:19) χ I k ( x λ ) ≤ µ + ( I k ) + 1 m (18)and µ − ( I k ) − m ≤ N N X λ =1 (cid:18) − ε λ (cid:19) χ I k ( x λ ) ≤ µ − ( I k ) + 1 m (19)where µ + ( I k ) = p (cid:0) k +1 m (cid:1) − p (cid:0) km (cid:1) , µ − ( I k ) = n (cid:0) k +1 m (cid:1) − n (cid:0) km (cid:1) and µ + = µ p , µ − = µ n arethe positive and negative variations of µ = µ ϕ (cf. the proof of Th.7).Now consider an arbitrary interval J = [ a, b ] ⊆ I , then there are subintervals J , J of J each one of them being a finite union of succesive intervals I k , such that J ⊆ J ⊆ J and µ + ( J ) − µ + ( J ) < ε, µ + ( J ) − µ + ( J ) < ε and µ − ( J ) − µ − ( J ) < ε, µ − ( J ) − µ − ( J ) < ε. (20)These inequalities are easy consequence of (17), i.e. of the uniform continuity of p and n . By adding at most m inequalities of the form (18), we get that µ + ( J ) − m ≤ N N X λ =1 (cid:18) ε λ (cid:19) χ J ( x λ ) ≤ N N X λ =1 (cid:18) ε λ (cid:19) χ J ( x λ ) ≤≤ N N X λ =1 (cid:18) ε λ (cid:19) χ J ( x λ ) ≤ µ + ( J ) + 1 m ;then using (20) we conclude that µ + ( J ) − ε < µ + ( J ) − m − ε ≤ N N X λ =1 (cid:18) ε λ (cid:19) χ J ( x λ ) ≤ µ + ( J )+ 1 m + ε < µ + ( J )+2 ε. (21)In a similar way we get that µ − ( J ) − ε < µ − ( J ) − m − ε ≤ N N X λ =1 (cid:18) − ε λ (cid:19) χ J ( x λ ) ≤ µ − ( J )+ 1 m + ε < µ − ( J )+2 ε. (22)Since (21) and (22) are independent of J , we conclude that lim N →∞ D N ( ω ; p ) = 0,lim N →∞ D N ( ω ; n ) = 0 and thus by (16) lim N →∞ D N ( ω ; ϕ ) = 0.22 emark 9. It is also possible (and useful) to define the concept of D ∗ N discrepancy forthe functions ϕ, υ, p and n (cf. Def. 1.2, p.90 of [15]). For instance we may define D ∗ N ( ω ; ϕ ) = sup ≤ x ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X k =1 ε k χ [0 ,x ) ( x k ) − ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It is then easy to see that D ∗ N ( ω ; ϕ ) ≤ D N ( ω ; ϕ ) ≤ D ∗ N ( ω ; ϕ ) (cf. Th. 1.3, p.91 of [15]). So we get that lim N →∞ D N ( ω ; ϕ ) = 0 ⇔ lim N →∞ D ∗ N ( ω ; ϕ ) = 0 . Let ( f n ) be a sequence of scalar valued functions defined on a set X . Given a strictlyincreasing sequence of positive integers 1 ≤ N < N < · · · < N k < . . . , we arrange theterms of ( f n ) setting g N = f , for 1 ≤ N < N and g N = f k , for N k − ≤ N < N k , k ≥ . Then the following Lemma has an easy proof, which we omit.
Lemma 7. If f n → f pointwise on X , then g N → f pointwise on X . Let now f n : N → R , n ≥ n ≥ lim m →∞ f n ( m ) = 0. We consider a strictly increasing sequence of positive integers( N k ) k ≥ such that m ≥ N k ⇒ | f k ( m ) | ≤ k , for k ≥ . (Since each f n is a null sequence of scalars, such a sequence exists). If we apply theabove arrangement to ( f n ) defined by ( N k ), we get the following Lemma 8. lim N →∞ g N ( N ) = 0 .Proof. Let N ≥ N , then there is k ≥ N k − ≤ N < N k , hence g N ( N ) = f k ( N ). Butsince N ≥ N k , we get (from the definition of ( N k )) that | f k ( N ) | ≤ k and so | g N ( N ) | = | f k ( N ) | ≤ k . As N → ∞ implies k → ∞ , we obtain the desired result.We are now in a position to prove the desired generalization of Theorems 7 and 8. Theorem 9.
Let ϕ : [ a, b ] → R be a function of bounded variation with ϕ ( a ) = 0 and V ba ϕ = 1 . Also let υ, p and n be the functions of total, positive and negative variationsof ϕ . Then there are sequences τ = ( x n ) ⊆ [ a, b ] and ε = ( ε n ) ⊆ {± } , such that forevery x ∈ [ a, b ] we have a ) lim N →∞ N P Nk =1 χ [ a,x ) ( x k ) = υ ( x )( b ) lim N →∞ N P Nk =1 ε k χ [ a,x ) ( x k ) = ϕ ( x ) , ( c ) lim N →∞ N P Nk =1 (1 + ε k ) χ [ a,x ) ( x k ) = p ( x ) and ( d ) lim N →∞ N P Nk =1 (1 − ε k ) χ [ a,x ) ( x k ) = n ( x ) . Proof.
We first reduce the theorem to the case when [ a, b ] is the unit interval I = [0 , g ( x ) = ( b − a ) x + a, x ∈ I ; clearly g is strictlyincreasing, with g (0) = a and g (1) = b . Set Φ = ϕ ◦ g and notice that(i) Φ is of bounded variation on I with Φ(0) = 0 and V Φ = V ba ϕ = 1 and(ii) υ Φ ( x ) = υ ϕ ( g ( x )) , p Φ ( x ) = p ϕ ( g ( x )) and n Φ ( x ) = n ϕ ( g ( x )), for x ∈ I . Now let( z n ) ⊆ I and ( ε n ) ⊆ {± } satisfying conditions (a) to (d) for the function Φ. Then thesequences x n = g ( z n ) , n ≥ ε n ) satisfy the same conditions for ϕ .In order to prove the theorem (with [ a, b ] = I ) we will follow the method of proof ofTheorem 8 (Th. 4.3, p.138 of [15]) and use Prop. 7. So let ϕ k , υ k (and p k , n k ) be asin Prop. 7. Since each υ k is continuous and increasing with υ k (0) = 0 and υ k (1) = 1,by Lemma 3 (Lemma 4.2, p.137 of [15]) there is a sequence τ k = ( x k , x k , . . . , x kn , . . . )satisfying (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X n =1 χ [0 ,x ) ( x kn ) − υ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ log ( N + 1) N · log N ≥ x ∈ I .It is clear that if we fix some k ∈ N , then letting N → ∞ we havelim N →∞ N N X n =1 χ [0 ,x ) ( x kn ) = υ k ( x ) for x ∈ I. (24)Let µ k = µ ϕ k be the Lebesgue-Stieljes measure that ϕ k defines on I and h k = dµ k d | µ k | bethe corresponding Radon-Nikodym derivative. Since each ϕ k is polygonal, its derivativeis a step function, hence h k is | µ k | -Riemann integrable, which implies by Prop.6 thatthe sequence ε k = ( ε kn ) n ≥ , where ε kn = h k ( x kn ) , n ≥ N →∞ N N X n =1 ε kn χ [0 ,x ) ( x kn ) = ϕ k ( x ) for x ∈ I. (25)It follows from Lemma 6 and Remark 9 that, if we set D ∗ N ( τ k ; ϕ k ) = sup ≤ x ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X n =1 ε kn χ [0 ,x ) ( x kn ) − ϕ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , then we have D ∗ N ( τ k ; ϕ k ) −→ N →∞ , for every k ≥ . (26)24e notice that we may furthermore assume that D ∗ N ( τ N ; ϕ N ) −→ N →∞ . (27)In order to attain (27), we consider a strictly increasing sequence of positive integers( N k ) k ≥ such that m ≥ N k ⇒ D ∗ m ( τ k ; ϕ k ) ≤ k for k ≥ . Then we arrange the sequence of functions ( ϕ k ) k ≥ as in Lemma 7, that is we setΦ N = ϕ , for 1 ≤ N < N and Φ N = ϕ k , for N k − ≤ N < N k , k ≥ . It then follows from Lemmas 7 and 8 that Φ k → ϕ , (Υ k → υ , etc.) pointwise on I andthat (23) to (27) remain valid for the sequence (Φ k ) k ≥ . So we may (and will) assumewithout loss of generality that Φ k = ϕ k , for k ≥ τ = ( x n ) ⊆ I by listing succesively the first term of τ , the first two terms of τ , . . . , the first k terms of τ k , that is, τ = ( x , x , x , . . . , x k , x k , . . . , x kk , . . . ) . The sequence of signs ε = ( ε n ) is constructed similarly; so we set ε = ( ε , ε , ε , . . . , ε k , ε k , . . . , ε kk , . . . ) . We are going to prove that τ and ε are the desired sequences. Assertion (a) of thistheorem is proved the same way as assertion (a) of Theorem 8.Indeed, by Lemma 4.1, p.136 of [15], it suffices to prove thatlim k →∞ k k X i =1 χ [0 ,x ) ( x ki ) = υ ( x ) , for x ∈ I. By using (23) and as (by Prop.7) υ k ( x ) → υ ( x ), for x ∈ I , we get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k k X i =1 χ [0 ,x ) ( x ki ) − υ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k k X i =1 χ [0 ,x ) ( x ki ) − υ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | υ k ( x ) − υ ( x ) | ≤≤ log ( k + 1) klog | υ k ( x ) − υ ( x ) | −→ k →∞ , for x ∈ I. This way assertion (a) is proved.Now, to prove assertion (b), using again Lemma 4.1 of [15] as above it suffices toshow that lim k →∞ k k X i =1 ε ki χ [0 ,x ) ( x ki ) = ϕ ( x ) , for x ∈ I.
25e now use (27) and the fact that ϕ k ( x ) → ϕ ( x ), for x ∈ I (see Prop.7), so we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k k X i =1 ε ki χ [0 ,x ) ( x ki ) − ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k k X i =1 ε ki χ [0 ,x ) ( x ki ) − ϕ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | ϕ k ( x ) − ϕ ( x ) | ≤≤ D ∗ k ( τ k ; ϕ k ) + | ϕ k ( x ) − ϕ ( x ) | −→ k →∞ . Assertions (c) and (d) follow easily from (a) and (b). The proof of the theorem is nowcomplete.
Concluding remarks
1. Let ϕ : [ a, b ] → R be a differentiable function with ( ϕ ( a ) = 0 and) bounded derivative.Then it is Lipschitz continuous and (hence) of bounded veriation. Let µ = µ ϕ be theLebesgue-Stieljes measure defined by ϕ on [ a, b ] and h = dµd | µ | . Assuming that ϕ ′ isRiemann integrable, it is not difficult to show that h is | µ | -Riemann integrable. It theneasily follows that if ϕ is piecewise C (for instance a polygonal function), then h hasthe desired property.On the other hand if ϕ ′ is not Riemann integrable, that is, ϕ is a Volterra typefunction, then the function h may or may not be | µ | -Riemann integrable. For examples(and the properties) of functions of Volterra type, we refer the reader to the books [11],pp.35-36 and [6], pp.22-25 and 33-35.2. Concerning future work, we note the following:(a) It would be interesting to have a generalization of Theorems 7 and 8 for functionsof ”bounded variation” of several variables, that is, for functions f defined on the cube I n for n ≥
2, which satisfy a proper notion of bounded variation (see for instance Def.5.2, p.147 of [15]).(b) Besides compact metric spaces (see Th.3), there are several classes of compact non-metrizable spaces K with the property that every measure µ ∈ P ( K ) admits a u.d.sequence (see [16] and [18]). For such spaces it would be interesting to know if everysigned measure µ with k µ k = 1 admits a u.d. sequence in the sense of Def.1 (cf. alsoRemark 5(b)). In our opinion the most interesting case is that of compact separablegroups G ; since we know that under special set-theoretic assumptions, i.e. ContinuumHypothesis (CH) (see [16]) or Martin’s axiom plus the negation of Continuum Hypothesis(MA+ ¬ CH), (see [8]), every measure µ ∈ P ( G ) admits a u.d. sequence. We note thatit is enough to consider the compact group { , } c , where c = the cardinality of thecontinuum; this is so because, as is well known, every compact separable group is adyadic space, i.e. a continuous image of { , } c (see [12] and [18]).(c) Regarding our consideration, the case of locally compact and separable metrizablespaces is also of interest, see [15] Notes, pp.177-178.26 eferences [1] F. Albiac and N.J. Kalton, Topics in Banach Space Theory, Grad. Texts in Math., vol.233, Springer, NewYork (2006).[2] S. Argyros and S. Mercourakis, On Weakly Lindel¨of Banach Spaces, Rocky Mountain J. Math. (1993),no. 2, 395–446.[3] S. Argyros, S. Mercourakis and A. Tsarpalias, Convex unconditionality and summability of weakly nullsequences, Israel J. Math. (1998), 157—193.[4] W.G. Bade, The Banach Space C(S), Lecture Notes Series no. 26, Matematisk Institut, Aarhus Universitet,Aarhus, 1971.[5] J. Bourgain, A. Pajor, S.J. Szarek and N. Tomczak-Jaegermann, On the duality problem for entropy numbersof operators, In Geometric Aspects of Functional Analysis, Lecture Notes in Math., , Springer, Berlin(1989), 50–63[6] Andrew Bruckner, Differentiation of Real functions, CRM Monograph Series, Amer. Math. Soc., 1994.[7] N.L. Carothers, Real Analysis, Cambridge Univ. Press, 2000.[8] R. Frankiewicz, G. Plebanek, On asymptotic density and uniformly distributed sequences, Studia Mathe-matica (1996), no. 1, 17–26.[9] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second Edition, J. Wiley,1999.[10] M. Fabian, P. Habala, P. H´ajek, V. Montesinos and V. Zizler, Banach Space Theory, The Basis for Linearand Nonlinear Analysis, CMS Books in Mathematics, Canadian Mathematical Society, Springer, 2011.[11] Russel A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Grad. Studies in Math., Amer.Math. Soc., 1994.[12] S. Grekas and S. Mercourakis, On the measure theoretic structure of compact groups, Trans. Amer. Math.Soc. (1998), 2779–2796.[13] Richard Haydon, Some more characterizations of Banach spaces containing ℓ , Math. Proc. CambridgePhilos. Soc. (1976), no. 2, 269-–276.[14] P. H´ajek, V. Montesinos Santalucia, J. Vanderwerff and V. Zizler, Biorthogonal systems in Banach spaces,CMS Books in Mathematics/ Ouvrages de Math´ematiques de la SMC, , Springer, New York, 2008.[15] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, J. Wiley, New York, 1974.[16] V. Losert, On the existence of uniformly distributed sequences in compact topological spaces I, Trans. Amer.Math. Soc. (1978), 463–471.[17] S. Mercourakis, On Cesaro summable sequences of continuous functions, Mathematika (1995), 87–104.[18] S. Mercourakis, Some remarks on countably determined measures and uniform distribution of sequences,Mh. Math. (1996), 79–111[19] H. Niederreiter, On the existence of uniformly distributed sequences in compact spaces, Compositio Math-ematica (1972), no. 1, 93–99.S.K.Mercourakis, G.VassiliadisUniversity of AthensDepartment of Mathematics15784 Athens, Greecee-mail: [email protected]@hotmail.com(1972), no. 1, 93–99.S.K.Mercourakis, G.VassiliadisUniversity of AthensDepartment of Mathematics15784 Athens, Greecee-mail: [email protected]@hotmail.com