On metric space valued functions of bounded essential variation Текст научной статьи по специальности «Математика»

Trudy Petrozavodskogo Gosudarstvennogo Universiteta

Seria “Matematika” Vipusk 12, 2005

V.I.K 517

ON METRIC SPACE VALUED FUNCTIONS OF BOUNDED ESSENTIAL VARIATION

M. Balcerzak, M. Malolepszy

Let 0 = T C M and let X be a metric space. For an ideal J C P(T) and a function f: T ^ X, we define the essential variation VJs(f,T) as the infimum of all variations V(g,T) where g: T ^ X g = f on T \ ^^d E e J. We show that if X is complete then the essential variation of f is equal to inf{V(f, T\E) :

E e J}. This extends former theorems of that type. We list some consequences that are analogues to the recent results by Chistyakov.

Some examples of different kinds of essential variation are also investigated.

Introduction

Let T be a nonempty subset of the real line. Let J be an ideal of subsets of ^^us J is a nonempty hereditary and additive subfamily of P(T) (the power set of T) with T </ J. An ideal is usually interpreted as a family of small sets that are negligible in the respective sense. Sometimes, one assumes additionally that J contains all singletons {t}, t e T, or/and it does not contain nonempty open sets in T. If an ideal is a-additive, it is called a a-ideal. The following families form well-known ideals on the real line: finite sets, nowhere dense sets, countable sets, Lebesgue null sets, sets of the first Baire category. The last three examples are a-ideals. (See [1].) These examples may produce ideals on T c M by taking intersections of TT

t e J).

Let N = {1,2,...} Fix a metric space X with a metric d, and let 0 = T c M. If x e M is a right (left) limit point of T then by f (x+),

© M. Balcerzak, M. Malolepszy, 2005

f (x—) we denote the onesided limits of a function f : T ^ X at x, provided that they exist. By Lip(T,X) we denote the set of all Lipschitz functions from T into X, and by Ld(f, T) we denote the smallest Lipschitz constant for f. A finite sequence T = {ti}n=0 such that t0 < ... <tn and ti e T for i = 0,..., n is called a partition of T. The Jordan variation of f : T ^ X is defined by

n

V (f, T) = sup {J2 d(f (ti), f (ti-i)): n e n} .

T i=i

(See [2], [3], [4].) If V (f,T) < ro, we say th at f is of bounded variation and we write f e BV(T,X). Consider an ideal J c P(T). The essential variation of f : T ^ X with respect to J is defined as the following quantity:

Vjss(f, T) = inf{V(g, T) : tee are E e J

and g: T ^ X such that g = f on T \ E}.

If Vjs(f, T) < ro, we say that f is a function of bounded essential variation with respect to J, ^d we write f e BVJss(T,X).

Essential variation was considered before in [5], [6], [7], [8], [2] but only for the ideal of Lebesgue null sets on the real line. In the present paper we generalize a characterization of essential variation given in [7] and [2]. Namely, we show that VJ£s(f, T) equals inf{V(f, T\E): E e J} (Theorem

BVe ss

those proved in [2] (Theorem 3). We show how one can extend a function BV(T \ E, X) T

variation (Theorem 1). A related result was given in [9] with another proof. Our paper contains general facts and examples witnessing a significant

Ve ss ( f, T) J

Characterization of essential variation

We are going to prove that a metric space valued function of bounded

TcM

T

earlier by Chistyakov and Rychlewicz [9, Theorem 1(a)]. In the proof given in [9], the authors apply a structural theorem for functions of bounded

variation [4, Theorem 3.1], [2, Lemma 2.1]. Our proof is different and seems more elementary.

We start with the following lemma.

Lemma 1. Let0 = D c M, X be a complete metric space, f e BV (D,X) and assume that t is a right-sided (left-sided) limit point of a set D. Then

ft

A proof is quite similar to that in [3, Theorem 1.24(e)],

0 = T c M E c T 0 = E = T

f: T \ E ^ X and f e BV(T \ E,X) where X is a complete metric space. Then there is a function h: T ^ X such that V(h, T) = V(f,T \ E) and h\T\E = f ■

Proof. Define h: T ^ X as follows. Let t e T. If t e E we put h(t) = f (t). Now 1 et t e E. In the ease t e E and (T \ E) n (—ro,t) = 0 we

denote t* = sup((T \ E) n (—ro,t)). If this supremum is attained, we

put h(t) = f (t*), and otherwise let h(t) = f(t* —) (this limit exists by Lemma 1). In the ease (T \ E) n (—ro,t) = 0, from E = T it follows that (T \ E) n (t, ro) = 0. Thus we denote t* = inf((T \ E) n (t, ro)). If this infimum is attained, we put h(t) = f (t*), and otherwise let h(t) = f (t*+) (this limit exists by Lemma 1).

Since h\T\E = f, we have V(f,T \ E) < V(h,T). Now, we will show the converse. For a partition T = {ti}”=^ of T we denote S(h, T) = Ylt==i d(h(ti), h(ti-i)). We will prove that for each partition T of T there T* T

S (h, T) < S (h, T*) < V (f,T \ E)

which yields the assertion.

So, let T = {ti}m=0 be a given partition of T. We modify T in four steps:

1) If to e E and (T \ E) n (—ro,t0) = 0, we insert a point from (T \ E) n (—ro,t0) to the partition.

2) If tm e E and (T \ E) n (tm, ro) = 0 we insert a point from (T \ E) n (tm, ro) to the partition.

3) For every pair ti,ti+i e E, if (ti,ti+i) n (T \ E) = 0, we insert a point from (ti,ti+i) n (T \ E) to the partition.

4) We look for all maximal strings ti,ti+i,..., ti+k with [ti,ti+k] n T c E. (Since E = T, by our modifications 1), 2), 3) it is impossible

h

[ti,ti+k] n T. We delete points ti+i, ...,ti+k from the partition. This S(h, T)

T*

3) can only enlarge S(h, T), so we have S(h, T) < S(h, T*). In partition T* T \ E

simplicity assume that T* = {ti}m=0■ By 1), 2) we have ensured that t0,tm eT \ E, if it is possible.

Now fix i e {0,..., m}. Consider three cases:

Case 1.2 < i < m and ti-2 e T \ E, ti-i e E, ti e T \ E. If (T \

E) n (—ro, ti-i) has a maximal element t*-i, then ti-2 < t*-i < ti-i and d(h(ti-2), h(ti-i))+d(h(ti-i), h(ti)) = d(f (t—), f (t*-i))+d( f (t—), f (ti)).

If (T\ E) n (—ro, ti—i) ^o maximal element, we set t*-i = sup(T\ E) n (—ro, ti-i). Of course ti-2 <t*-i < ti-i. Pick a sequence {t*-i,n}^Li of numbers from (T\ E) n (—ro,ti-i) such that limn^TO t*-i n = t*-i. We have

d(h(ti—2), h(ti-i)) + d(h(ti-i), h(ti)) =

= d(f (ti-2), h(ti-i)) + d(h(ti-i), f (ti)) =

= lim [d(f (ti-2 ),f (t*-i,n)) + d(f (t*-i,n),f (ti))] < n—’ ’

< V(f, [ti-2,ti] n (T \ E)).

The last equality results from Lemma 1.

t0 E ti T \ E (T \ E) n (t0, ro) element t0*, then t0 < t0* < ti and d(h(t0),h(ti)) = d(f(t0*),f(ti)). If (T\ E) n (t0, ro) has ^o minimal element, we set t0* = inf(T\ E) n (t0, ro) and of course t0 < t0* < t^ Pick a sequence {t0*n}^=i of numbers from (T \ E) n (t0, ro) such that limn—TO t0*,n = t0*. We have

d(h(t0), h(ti)) = d(h(t0), f (ti)) =

= lim d(f(t0*,n),f(ti)) < n—

< V(f, [t0,ti] n (T \ E)).

The last equality results from Lemma 1.

Case 3. Let tm-i e T \ E, tm e E. If (T \ E) n (—ro,tm) has a maximal element t^, then tm-i < t*m < tm and d(h(tm-i),h(tm)) =

d(f (tm-i), f (t^)). If (T \ E) n (—ro, tm) has no maximal element, we set tm = sup(T \ E) n (—ro, tm). Of course tm-i <t*m < tm. Pick a sequence {t*m,n}£= °f numbers from (T \ E) n (—ro, tm) with limn—TO t*m,n = t*m-We have

d(h(tm—1), h(tm)) = d(f (tm—1), h(tm)) =

= limn—TO d(f (tm-i), f (t*m,n)) <

< V( f, [tm-i,tm] n (T \ E)).

The last equality results from Lemma 1.

Taking into account cases 1), 2), 3), by adding the respective sides of inequalities we obtain

S(h, T*) < V(f, [t0, tm] n (T \ E)) < V(f, T \ E).

The theorem has been proved. □

Now we are ready to prove our main result, a characterization of essential variation. In the case when T = [a,b], X = ^^d J is the

T

and El-Sayed in [7j. If T is a density-open subset of ^aid J is a family

TX

f e BVJss(T,X), the result was obtained by Chistyakov in [2]. In our paper we consider a more general situation where essential variation is

T

on the real line.

Theorem 2. Let X be a complete metric space and 0 = T c M and let J be a proper ideal of subsets of T. Then for every function f: T ^ X we have

Vjss(f,T) =inf{V(f,T \ E): Ee J}.

Proof. (Compare with [2, Theorem 2.1].). Denote v = inf{V(f, T \ E) : E e J}. If v = ro, then V(f,T \ E) = ro for all E e J. Hence for each function g: T ^ X such that f \T\E = g\T\E with E e J, we have

V(g, T) > V(g, T \ E) = V(f, T \ E) = ro.

Thus V£s(f,T) = ro Now assume that v < ro. For a fixed e > 0 there exists a set E0 e J such that V(f,T \ E0) < v + e. By Theorem 1 we find

a function g0 e BV(T,X) such that f \T\Eo = g0\T\Eo and V(g0,T) = V (f,T \ E0). Then

VPJss(f, T) < V(g0, T) = V(f, T \ E0) < v + e.

Consequently, VJss(f, T) < v. It suffices to show the reverse inequality. By definition of VJss(f,T), for any number a > VJss(f,T) we find a function gi:T ^ X and a set Ei e J such that V(gi,T) < a and gi\T\El = f \t\Ei- We have

V (f, T \ Ei) = V (gi, T \ Ei) < V (gi,T) < a.

It follows that v < V (f,T \ Ei) < a and since a > VJss (f, T) is arbitrary, we obtain v < VJss(f,T). □

Some examples and properties

Let us start with a simple observation when one considers two ideals T

0 = T c M X

that I, J c P(T) are ideals and f: T ^ X. Then we have:

(a) if I c J then Vlss(f,T) > V£s(f,T),

(b) V?sflJ(f,T) > max{^s(f,T),V£s(f,T)}.

Proof. Assertion (a) is an immediate corollary from the definition of

□

Now, we will to show that the essential variation in one sense can be small and in another sense - can be large.

Proposition 2. Let 0 = T c M and let I, J c P(T) be two ideals such that there exist a set A e I\J and a strictly monotonic sequence (xn)TO=0 of numbers from T such that An e J for all n e N where An = A n \xnin{xn-i,xn}, max{xn-i,xn}), n e N. Then for every complete metric space X of cardinality > 2 there exists a function f: T ^ X with

Vlss(f,T)=0and VeJss(f,T) = ro.

Proof. Let A be ^ in the assumption. Pick distinct points x,y e X. Define f: T ^ X by putting f (t) = x if t e UTO=i A2n and f (t) = y for t e T \ UTO=i A2n- ^ce A e I and f \T\^ is constant, by Theorem 2 we

obtain V^ss(T, X) = 0. To show that VJs(T, X) = ro, fix E e J. We have An \ E e J for each n e N, thus we can pick tn-i e An \ E, n e N. The sequence (tn)TO=0 is strictly monotonic. Its beginning part (tk)n=0> n e N,

T\E

numeration). Consequently,

n

V(f,T \ E) > supV d(f (tk),f (tk-i)) = sup nd(x,y) = ro.

nGN k==i nGN

Hence by Theorem 2 we obtain VJss(f,T) = ro □

I, J c P(T)

sets A,B c T such that A e I, B e J and A U B = T. The a-ideals of

M

T=M

of orthogonal ideals, see for instance [10].

T M I, J c P( T)

T

X> f: t ^ X with Vlss(f, T) = 0 and VJ = ro.

T

(yn)TO=0 °f numbers from T. Put xn = y2n for n e N U {0}. Let A,B c T be such that A e I, B e J, A n B = 0, A U B = T. It suffices to show that An n N J

instance that (yn)TO=0 is increasing. Suppose that Ak e J for some k e N. Observe that U = (xk-i, xk) n T is open in T and U = 0 since y2k-i e U. Because Ak e J,w have A n U = Ak n U e ^^om B e J it follows that B n U e J. Consequently U = (U n A) U (U n B) e J which yields

□

Example. Let I be the ideal of all sub sets of [0,1] of Lebesgue measure zero and let J be the ideal of all sub sets of [0,1] of the first category. It is well known that I and J are two orthogonal a-ideals, so pick two disjoint sets A Mid B such that A e I and B e J and [0,1] = A U B. Since A is residual, its intersection with every nondegenerate interval is of the second category. Let An = [1 — n, 1 — -L_) n A for n e N. Define f: [0,1] ^ M by putting f (t) = 0 if t e B U {1^ rnd f (t) = 2^ if t e An. Since A e I and f\[0ji]\A is constant, by Theorem 2 we obtain V^^f, [0,1]) = 0. Since An E J n N

have An \ E = 0. Hence VJs(f, [0,1]) = Y,^=i() = 2- For eacl1 E e I n J and for every interval [a, b] c [0,1] there exist t,s e [a, b] such that t e B \ E and s e A \ E. Hence V^J(f, [0,1]) = ro. This shows that V1^J(f, [0,1]) > max^sf, [0,1]), VJs(f, [0,1])}. Hence in the assertion (b) of Proposition 1, we cannot use equality.

Let X be a metric space. We say that A c X is precompact in X if the closure A is compact. The following theorem collects some consequences of Theorem 2 analogous to those presented in [2].

0 = T c M X f :

T ^ X.Let J be a a-ideal of subsets ofT. Then

(a) f e BVJss(T,X) if and only if there exists a set E e J such that f \T\E e BV(T\E, X); moreover, E can be chosen such that V( f, T\

E) = Vjss(f,T).

(b) If {fn}^=i c BVJ(T,X) and d(fn(t), f (t)) ^ 0 as n ^ ro for teT \ E, where E e J, then V£s(f, T) < liminf n^^ V£s(fn, T).

^) (Structural Theorem) f e BVJss(T,X) if and only if there exists a nondecreasing bounded function ^ : T ^ M and a function g e Lip(D, X), where D = y>(T) and Ld(g, D) < 1, such that f = g o T \ E E J

(d) (Helly’s type Theorem) If F = {fn}™=i c BVJs (T, X),

sup Vjss(fn,T )

n£N

is Unite and the set {fn (t)}JJ=i is precompact in X for teT \ E, where E e J, then F contains a subsequence which converges in metric d on T \ E to a function from BVJss (T, X).

With application of Theorem 2, the proof of Theorem 3 goes similarly as for [2, Theorem 2.2]. Thus we omit it.

Ja

unions) is essential in Theorem 3(a). Indeed, put E = {—: n e N} and

define f : [0,1] ^ M by putting f (t) = t if t e E and f (t) = 0 if t e [0,1] \ E. We have V(f, [0,1]) = n Consider as J the ideal of all

Unite subsets of [0,1]. We put En = {nm: m e N, m < n}, n e N. Thus

En e J. For all n e N we dehne gn: [0,1] ^ M by putting gn(t) = t if t e E \ En and gn(t) = 0 for the remaining t in [0,1]. Hence gn(t) = f (t)

for all n e N and t e [0,1] \ En, and V(gn, [0,1]) = 2 • = __L_ for

all n e N. Consequently, lim V(gn, [0,1]) = 0 and VeJs(f, [0,1]) = 0.

n—

However, for all D e J we have V(f, [0,1] \ D) > 0.

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Institute of Mathematics, Lodz Technical University, ul. Wolczanska 215, 93-005 Lodz, Poland E-mail: [email protected]

Center of Mathematics and Physics, Lodz Technical University,

al. Politechniki 11, 90-924 Lodz, Poland E-mail: [email protected]