ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXX (1990)
F
r a n c is z e kP
r u s- W
is n io w s k i(Szczecin)
Some remarks on functions of bounded (^-variation
Abstract. A characterisation of regulated functions and new proofs of the well-known theorems concerning the structure of regulated functions are given. It is shown that the intersection of all modular classes Vv is the linear space of regulated functions whose sets of values are finite.
Necessary and sufficient conditions are determined for a function to be of bounded (^-variation and some consequences of those conditions are proved. Some properties of well-regulated functions of bounded ^-variation for a superadditive (^-function are given.
Introduction. The notion of (p-variation was introduced by^L. C. Young in [17] and has various applications, for example to the study of convergence of Fourier series, ^-variation is an important example of a modular (see [10]), and in this paper some remarks about functions of bounded (p-variation are given relevant to the theory of modular spaces.
In the first section a slightly generalized definition of <p-variation is introduced and two fundamental lemmas are proved. The new definition enables us to consider the <p-variation of a function defined on an arbitrary set of real numbers. The second section contains new proofs of two well-known theorems about the structure of regulated functions ([13], Theorem 10 and [1], Theorem 1). Both proofs are based on a characterisation of regulated functions given in our Lemma 2.1. The third section concerns the intersection of function classes of bounded (p-variation over all (p-functions. In the fourth section some necessary and sufficient conditions are proved for a function to be of bounded
^-variation. Moreover, it is shown that for modular classes of functions of bounded <p-variation an equality of Lesniewicz type holds (cf. [7], equality II).
The last, fifth section contains some facts similar to Theorem 1 and Theorem 2 from [14].
0. Preliminaries
0.1. (p-functions. A function <p: [0, oo)-> [0, oo) is said to be a (p-function if
it is continuous, nondecreasing, <p(0) = 0, <p(n) > 0 for и > 0 and (p(n)-> оо for
w-MX). We denote by Ф the set of all (p-functions. Two (p-functions <p, ф are
said to be equivalent for small u, in symbols <p ~ ф, if there exist constants
148
F. P r u s - W i s n i o w s k ialf bl5 a2, b2, u0 > 0 such that a1 <;р(Ь1м) ^ ф{и) ^ a2cp(b2u) for и < и0. We set Ф + = {(реФ: ср is convex and lim cp{u)/u = 0},
и-*
o +Ф _
= [среФ: (p is concave and lim (p(u)/u =
o o }.u-+0 +
We say that (реФ satisfies the A2-condition for small и if limsupu_>0 + (p{2u)/q){u) <
oo.We denote by ФЛг the set of all (^-functions satisfying the A 2-conditior| for small u. The inclusion Ф_ <= фЛ2 holds (cf. [9], 1.63).
We will write qxuj/ (resp. q> <ф), whenever lin^up,,-^* (р(си)/ф(и) < oo for every c > 0 (resp. limu_>0 + (р{си)/ф(и) = 0 for every c > 0).
0.2. Л-variation. A nonincreasing sequence Л = (A„) of positive numbers is said to be a Л-sequence if An-+0 and YnK = 00 • A real function x defined on a closed interval [a, b] is said to be of bounded A-variation, in symbols
x e A - B V ,
if for all sequences
( [ a „ ,b j) of nonoverlapping subintervals of [a, b]
we have £„A„|x(bJ — x(a„)|
< oo.The A-variation of x is the supremum of Y„ >Ux(b„) — x(a„)| over all sequences ([a„, b„]) of nonoverlapping subintervals of [a, b].
0.3. Regulated functions. A real function x defined on a closed interval [a, b] is said to be regulated if it has a right- and left-hand limit at every point of la, b]. Regulated functions are bounded. For a regulated function x we denote by x(t + ) and x(f — ) the right-hand and the left-hand limit of x at a point t of [a, b] respectively. We write x{a — ) = x(a) and x(b-l-) = x(b). Since the set of discontinuities of every regulated function is countable, it follows from ([8], p. 71) that regulated functions are of first Baire class. We will denote by R the set of all real numbers and by R0[u, b] the linear space of all regulated functions defined on [a, b] which vanish at a.
1. (^-variation. Definition and fundamental lemmas. A finite subset n of A with the natural order will be called a partition of the set A c R. In general, if n is nonempty then we write the partition к in the form of a finite increasing sequence (£.)"= i. We denote by ПА the set of all partitions of A.
Given a (^-function q>, a real function x defined on A c R and a partition л of A, let us set
var^x; n) = 0 n 1
E <p(|x(ti+l)-x(£;)|)
i = 1
if card7r ^ 1, if 7 1 = (t{)"=i, n ^ 2.
Then var(p(x; n) will be called the cp-variation of x with respect to the partition ж and var^(x; A) = sup{var^(x; ж): жеПА} will be called the (p-variation of x on A. If var(p(x; A) is finite then we say that x is of bounded (p-variation on A.
It follows from the very definition that if В cr A then varv(x; В) ^ var^x; A).
Remarks on functions o f bounded cp-variation 149
This property is called the monotonicity of (^-variation with respect to set inclusion. If A is a closed interval [a, b], the above definition is equivalent to the classical one ([17], p. 582). Functions of bounded ^-variation are bounded.
Given two nonempty subsets A, В of R, we write A ^ В whenever sup A ^ inf B. We assume that 0 ^ A ^ 0 for every A a R.
1.1. L
e m m a. Assume that the set A c= R is nonempty. Then for a function x : A-+R, a qy-function q> and a sequence (AJ of subsets of A such that i ф j implies either A { ^ Aj or Aj ^ A t,
^ i) < v a r v(x; A).
i
Moreover, if x is bounded and (Jr=i^£ = ^ then
m
var^x; A )< £ var„(x; А ) + { т - \ ) ( р { о 8 с х А ) .
i — 1
P roof. We start with the proof of the first part of the lemma. It suffices to prove that var^x; A) ^ £"= t var(p(x; A ) for every n = 2 ,3 ,... We use induc
tion on n. Let A lf A2 cz A and Ax ^ A 2. Taking partitions n1 = {ti)1=1eIJAl and n2 = (si)il=1 е П А2, we have л 1'и л 2е.ПА and
var^x; Ti1) + var9,(x; n2) ^ var^x; 7i1) + <j[>'(|x(s1) - x ( t ll)|) + var^(x; n2)
■= varv(x; и n2) ^ var^x; A).
Thus
var^x; A ^ + var^x; A 2) ^ var<p(x; A).
Now, assume that for some integer n ^ 2 and for every positive integer к < n, every set B a R, every function y : B-+R, every ф-function q> and every sequence (B) of subsets of В such that i ф j implies either Bt ^ B j o r B ^ Bt we
have к
Z vaM y; Bi) < varv(y; B)
i = 1
Let A t, i = 1, ..., n + 1, be subsets of A such that either Ai < A j от Aj < Af for i Фj. Setting C1 = { ie { l, n}: A{ ^ A„ + 1} and C2 = {1, ..., n}\Cj, we have
( + ) U A, Л„+1 ^ U A,.
ieC i ieC 2
Because card Ct ^ n and card C2 < n, by the inductive assumption Z var^x; A j ) ^ var„(x;.(J Aj for j = 1,2.
ieC j ieCj
So by ( + ) and again by the inductive assumption we have
150
F. P r u s - W i s n i o w s k in + 1
Z var«>(*; Ai) ^ var^x; U ^ J + var^x; ^n + ^ + var^fx; (J A t)
i = 1 ieC i ieC 2
^ var^fx; An + 1 u (J ^ J + var^x; (J Л,.)
ieCi ieC2
л + 1
^ var„(x; IJ At) < var„(x; Л).
i = 1
Because the proof of the second part is similar, we consider only the case m = 2. Given two nonempty subsets A 1, A 2 of A such that A xkj A 2 — A and A x ^ A2 and taking a partition n = (ti)”=1e n A\ ( n Al и ПЛ2), we have n ^ 2 and tx$ A 2, t„$Ax. Setting
j = m ax{ie{l, n } : t ^ A ^ , к = min ( i e {1, ..., n } : t ^ A - , } , we obtain j ^ k ^ j + l and = {ti)j= 1 e n A i , я 2 = ( ^ =4б Я А2. Thus
*/ j - 1 n - 1
varv(x; я) = £ </>(|x(ti + 1) - x ( t £)|)+ J < p (M*» + i ) - x (* j )I)
i = l i = j ^ \
j — 1 n — 1
= X <p(|x(ri+i) — x(ti)l) + ф(|х(Гл) — x(f7-)l)H- Z <p(lx(*i + i)-*(*i)l)
i= 1 i = k
^var„(x; 7r1) + var(p(x; Ti2) + (p(oscx^).
This implies that
var<p(x; ^ ^ v a r ^ x ; A x) + var^(x; A 2) + (p(oscxA). •
1.2. L
e m m a. Let (u„) be a sequence of positive numbers convergent to 0 and let A be a countable infinite subset of (a, b). Then there exists x e R 0 [a, b] such that {te[a, b]: x(t) ф 0} = A and for every (p-function cp and every A ^ 0,
var^Ax; [a, b]) = 2 Z q>(Xun).
P roof. Arranging all elements of A in a sequence (r„), we define x: [а,Ь ]-> Я as follows:
x(0 = 0
for t = rn, 1 , 2 , . . . , for other t.
Clearly {te[a, b]: x(t) Ф 0} = A. Moreover, since (w„) converges to 0, we have x (tT ) = x(t — ) = 0 for every te [a , b]. Therefore, x is regulated.
Given a (^-function q> and A > 0, for a positive integer к we arrange the points rx, ..., rk in increasing order z x, ..., zk. For the partition nk = (rf)?*!1 such that tk2i = zi for i = l , . . . , f c , t*t = a, tk2k + l =b, tk2i- 1e{zi_ x, z{)\A for i = 2 ,..., к we have
к
var„(Ax; ?rfc) = 2 £ <р(Аи„).
n= 1
Remarks on functions o f bounded (p-variation 151
Thus
var^/bc; [ a , b]) ^ sup{var„(Ax; n k): к = 1, 2, ...} = 2£<р(Ами).
П
Now, we shall show the opposite inequality. Observe that for every t , s e [a, b]
with t < s and for every q e { t , s)\y4 we have |x(t) — x(s)| ^ max{|x(f)|, |x(s)|}.
Thus
( + ) q>(\x(s)-x(t)\) ^ <p(|x(g)-x(0|) + <p(|x(s)-x( 4 )|).
Taking an arbitrary partition n: a = tx < ... < tm = b, we set J = {ie {1, ..., m} : tt e A] and denote by nt (for i e I) the index of tt in the sequence (r„).
For the partition n0 = such that s 2 i- 1 = tf for i = 1 , m and s2ie(£i5 ti+i)\A f°r i = l , . . . , m —1 we have by ( + )
m — 1 2m — 2
X (p(X\x(ti+1)-x(ti)\) ^ X ç>(A|x(si + 1) - x ( s f)|) = 2^(p(hc(ti))
i = 1 i = 1 ig J
= 2£<jf>(Ax(rJ) ^ 2 X < p (^ jc (0 ) = 2 X
îg J и=1 n = 1
where к = шах{пг: г e /} (m ax0 = O). Thus var^Ax; n) < 2 X„<p(Au„) and finally var„(Ax; [a, b]) ^ 2 X„<p(Aw„). ■
2. The structure of regulated functions. Given a bounded function x: [a, b] a positive integer m and a partition л; = (tf)"=1 g Я [a, b], we set
&*,*(»») = card{ie (1, и- l } : |x(ti + 1) - x ( t f)|e(2_woscx, 2_w + 1oscJ}, M m) = s u p { k XfJC(m): тте Л [а, b]},
where oscx = oscx[a, b],
2.1. L
e m m a. A b o u n d e d f u n c t i o n x: [a, Ь]->Я is r e g u l a t e d i f a n d o n l y i f k x ( m ) is /mite /or снегу p o s i t i v e i n t e g e r m.
P ro o f. Suppose that x is not regulated. Then there exists a point s0e[a, b] at which x does not have one of the one-sided limits, say the right- -hand one. Clearly, s0 < b. Setting c = liminfs->s+x(s) and d = limsups^s+ x(s), we have c < d. Let m be the smallest positive integer such that 3oscx ^ 2m(d — c). Then there exists a decreasing sequence (sk) in [a, b] convergent to s0 such that
\ x { s 2 k - i )
— d\ < 2 ~ mo s c x and |x(s2fc) —c)| < 2 _moscJC for к = 1 ,2 ,...
Hence |x(sk+1) — x(sfc)| > 2_mosc for к = 1 ,2 ,... For an integer к > 0 we define the partition n k = e П [ а , b], setting t \ = sfc + 2_f. Then
m m
X kx(n) > sup{ X £х>Як(и): к =
1,2,...} = sup{/c: к =
1,2,...} = со.
n=i и=1
Thus кх(и) = oo for some и = 1 ,..., m.
Conversely, assume that x is regulated. Given a positive integer m, from
152
F. P r u s - W i s n i o w s k iour assumption it follows that for every se [a , b] there exists an es > 0 such that
sup{|x(t1) —x(t2)|: tl9 f2e(s —es, s ) n [ a , b]} < 2~mosc;c, s u p ^ x ^ ) — x(f2)|: tl5 t2e(s, s + ss) n [a, b]} < 2_mosc-x;.
Taking for every se [a, b] the open neighbourhood Us = (s — es, s + es) n [a, b], we get an open cover of [a, b] which contains a finite subcover t/Sl, , USn.
Observe that for every s e [a , b] and every partition (ti)li=l G n Us we have 2 ^ card [ie { 1 ,..., k — 1}: |x(ti+1) — x(£j| > 2 -moscx}.
Since [a, b] = (J"=1 USJ, for every partition n = ( t ^ i e ïï[ a , b] we have
= c a rd { ie { l,..., к - l } : I x f o - ^ - x ^ J I E(2~moscx, 2~m + 1oscJ and V/ [tj, ti+ i]^
+ c a rd { ie { l,..., к — 1}: |x(£i+1) — x(t;)|e(2_mosc:c, 2_m + 1osc;c]
and 3j [£f, tf+1] <= UsJ]
< c a r d { ie { l,..., fc-1}: V/ [tit ti + J 4 Usj}
П
+ X c a rd { ie { l,..., fc-1}: [tf, ti + 1] c I7s;
j= i
and |x(ti+1)-x (£ f)| > 2_moscx)
< n + 2n.
Hence kx(m) < oo. ■
2.2. P
r o p o s i t i o n. A function X : [a, b] -> Æ is regulated if and only if x is of bounded Л-variation for some А-sequence A.
P ro o f. First, assume that x: [a, b] is not regulated. If x is unbound
ed then хф А-BV for every Л-sequence A ([15], p. 108). If x is bounded then by Lemma 2.1, kx(m) = oo for some positive integer m. Thus, for every к = 1 ,2 ,...
there exists a partition nk of [a, b] such that kx<nk(m) ^ k. This implies that for every к = 1,2, ... there exists a finite collection {[a*, bf]: i = 1, ..., k} of nonoverlapping subintervals of [a, b] such that |x(b*) — x(a-)| > 2~moscx > 0 for i = 1, ..., k. Hence for every Л-sequence Л = (AJ and every integer к > 0 the A -variation of x is greater than ^■ = 12i2“mosc;c. Thus, by ([16], Theorem 1), хфА-BV for every Л-sequence Л.
Conversely,, suppose that x is regulated. Then x is bounded and by Lemma 2.1, kx(m) is finite for all m. If there exists a positive integer N such that kx(m) = 0 for m > N, then for an arbitrary Л-sequence Л = (AJ and arbitrary sequence ([af, bj) of nonoverlapping subintervals of [a, b] we have
ХА;|х(Ь;) - х ( а г)| ^ A1J]|x(bi) - x ( a i)| ^ At £ kx(m)2~m + l
i i i = 1
O SC v < OO .
Remarks on functions o f bounded (p-variation 153
Thus,
x eA - B V .Now, assume that X™MW) = 00• Let b0 — m0 = t0 = 0 and bx = kx(l), ml = 1. Setting for n = 1,2, ...
r m„ + 1
m„ + 1 = m in{re{l, 2, ...}: X MO ^ 2b„} and b„ + 1 = X MO,
i = 1 + m„ i = 1 + m„
we have ( + ) Setting we obtain
m „ ^ n and bn + 1 ^ 2nbl ^ 2n.
w„ = 2n/(n2b„) for n = 1 ,2 ,..., n V 2b.
1 2
( + + ) 0 < w„ < --2 and w„ + 1 = ^ - . ,
и n bn\n + l j bn + 1 < W„.
Let tn = X?= l bt = ХГ = 1 MO for n = 1,2, ... and A£ = wn for и = 1, 2, ... and i = ..., t„. Then by (+ + ) the sequence Л = (Аг) is a nonincreasing sequence of positive numbers tending to 0 and
X ^ = Z Ê wn = Y,bnwn = YJ\ = °o.
i n i = 1 + t„ - 1 n n n
This means that Л is a Л-sequence.
Let {[ui? b j: i = 1, ..., JV} be a finite collection of nonverlapping subin
tervals of [a, b]. By ([2], Theorem 368) we may assume without loss of generality that |x(b£) — х(<я£)| ^ |x(bi + 1) — x(a£+1)| for i = 1, ..., N — 1. Thus, if lx(b£) — x(a£)[ > 2_soscx then i ^ X™=iMm)- So if i > tn- 1 then by ( + )
|x(b£) - x ( a £)| ^ 2 m"~1 oscx ^ 2 " + 1oscx.
Finally, we have
X *i\x(bi)-x{aù I < X Ê ^.2“ " + 1oscx = oscxX X 2"" + 1w,
i = 1 n i = l + t n - i n i = l + t „ - i
= o s c X 2 " +1 b„w„ = oscx X 2 n 2 < oo.
n n
Thus, by ([16], Theorem 1),
x eA - B Vfor A = (A,). ■
Using the integral representation of a convex function (see [6], Theorem LI) we can easily prove the following lemma.
2.3. L
e m m a. Let q> be a (p-function. Assume that there exists a decreasing
sequence ( un ) convergent to 0 such that (p is linear on [ul , oo) and on every
[u„ + 1, unl Let а г denote the angular coefficient of cp on [wl5 oo) and for
n = 2, 3, ... let an denote the angular coefficient of cp on [un, «„_!]. Then (p is
154
F. P r u s - W i s n i o w s k iconvex if and only if an + 1 ^ an for every n — 1 ,2 ,...
2.4. Pr o p o s it io n.
A function
x : [a ,fr]
- + Ris regulated if and only if x is of bounded (p-variation for some <реФ+.
P ro o f. It is known that if x is of bounded (^-variation for some ^-function (p then x is regulated (see [17], (2.4)).
Conversely, suppose that x is regulated. Then by Lemma 2.1 we have 0 < kx(m) < o o for m = 1 ,2 ,... We define qy. [0, oo) —► [0, oo ) by
un = 2- " + 1osc:c for n — 1, 2 ,..., (p{0) = 0, (piuj = 2~1(1 + kx{\))~1,
(p{un) = min{2_1 cpiun-i), 2~nun, 2~"(1 +/cx(n))-1} for n = 2 ,3 ,...
Finally, we define <p(u) to be linear on [w2, oo) and on each [un+l, wj. It is easy to check that (p is a ^-function. By definition of (p for every positive integer n we have
2 - 2
" 1oscx(p(ufh+1) ^
2 " 1o s c x ç> (mJ ,and therefore, 4
<P(un + 1)(un- u n + 2)-(p(un){un + 1- u n + 2) ^ 0.
Hence, for every positive integer n
(P(u„ + i)-(p{un + 2) ^ 2 1q>(ut)-q>(un + 2)
n;
tin + 1 t l n + 2 t l n j. i Un + 2
n + l t l n + 2 / ч , ч
--- <p{un)-(p{un + 2) Hn Un + 2
u n + 1 u n + 2
<РЫ < <p(u„) t (p(un + 1){un- u n +
2)-(p{un){un + l - u n + 2)
t l n t l n + 2 t l n Un + 2 {Un W„-|_2 )( U n t l n + i )
= (p(un)-(p(un+ 1 )
Un ~ Un + l
Thus, by Lemma 2.3, (p is convex. For n = 2, 3 ,... and we[w„ + 1, u j we have л cp(u) (p(u„) 2~"w„
Q < T \ / ^ ' Г У П / ^ ___________n _ 2 ~ n + l
w an + j wn + 1
hence (реФ+.
Finally, for an arbitrary partition (^)Г=1 e/7 [a, b] we have
m_1 к In)
Z <jf>(|x(fi+1) —x(tf)|) ^ Y , k x(n)(p{2-n + 1oscx) ^ Z 2 ~" t Л / ч < !•
i = 1 n n
This proves that var^(x; [a, b]) ^ 1. ■
Let 2f denote the linear space of real functions defined on a closed interval
Remarks on fanerions o f bounded (p-variation
155[a, b] which vanish at a. For a ^-function <p the functional var^,: X ->[0, oo]
defined by var^Cx) = varç)(x; [a, b]) is a modular ([11], pp. 57-58). We write Vf = { x e l : V2 > 0 varv(Ax) < oo}, = {x eX : varv(x) < oo}
and V* = { x e X : 3X > 0 var^/bc) < oo} (cf. [12], pp. 12-13). Vf is the greatest linear space contained in Vv, and V* is the linear space spanned in
* by V9.
2.5. L
e m m a. For every среФ+ there exists феФ + such that ф<ир.
Proof. Setting ф(и) = (p(u2) for w ^ 0, it is easy to see that феФ+. Given c > 0, we have ф(си) = q>(c2uu) ^ (p(u) for и ^ 1/c2. This proves that ф<ир. ■
Now, we are in a position to prove the following result.
2.6. P
r o p o s i t i o n. The following identities hold:
U K = U K = { J V* = R0la,ti].
<реФ + (реФ + (реФ +
P roof. By Proposition 2.4 we have [](реФ+ К? = Я0[а, b]. So it is enough to show that ( j ^ + V* a Vf, because Vf c F ? c V*. But this follows immediately from 2.5, because Vf a Vf iff ф<кр ([4], 4.4). ■
2.7. C
o r o l l a r y. The following identities hold:
U K° = U K = U V* = R0la, Ь].
<реФ <реФ <реФ
2.8. Ы
е м м а. For a (p-function <р and a positive integer п there exists a decreasing sequence (uk) of positive numbers convergent to 0 such that ux ^ 2~"
and Y s k V i ™ 1 uk) — oo fo r m = 1 , 2 , . . .
P ro o f. Given a (^-function q> and an integer n > 0, we set
S — 1
as = m in{ke{l, 2, ...}: k(p(s~l 2~n~s) ^ 1} and ts = Z at fo rs = 1, 2, ...
i = 1
Taking a decreasing sequence (ufc) such that uke{2~n~s, 2~n~s+l~\ for k = ts+ 1, ..., ts + as and s — 1, 2, ..., we have u1 ^ 2~n, limkuk = 0 and
+
1
COts + 1
Z<p(m~4) = Z Z (P{m'l uk) ^ Z Z <P(S~ 4 )
k s fe = 1 + ts
s
= m fc= l+ rsoo ts+ 1
S* I X ^(s-‘2-"“s)
s — m k — 1 -Vts
O O 00
= Z as<p(s_12~"~s) ^ Z 1 = 00
s=m s=m
for m = 1 ,2 ,... ■
2.9. P
r o p o s it io n. For every sequence (<p„) of cp-functions the proper
156
F. P r u s - W i s n i o w s k iinclusion [j„V*n
c=R 0[a,
b ]holds.
Proof. Given a sequence (cp„) of «^-functions, according to Lemma 2.8 for every integer n > 0 there exists a decreasing sequence ( m £) of positive numbers convergent to 0 such that и" ^ 2~n and £ k(p„(m_1u£) =
oofor m = 1, 2 ,...
Arranging the numbers ul (k, n = 1 ,2 ,...) in nonincreasing order, we get a sequence (uk) convergent to 0 such that ]Tk(pn(m-1 uk) ^ £ k(/>n(m-1 uk) =
oofor any positive integers n, m. Thus in view of 1.2 there exists x e R 0[a, b] such that хф {]пУ*п. ш
2.10.
Le m m a.I f (реФА2 then there exist r, d > 0 such that <p{2~n) ^ dr~n for n = 1,2, ...
P ro o f. If (реФА2 then r = sup{(p(2u)/(p(u): ие{0, 2-1]} is finite ([12], 1.02). Setting d — q>{ 1), by an induction argument we have <p(2“" + 1) < r(p(2~n) for n = 1, 2, ... and thus d ^ rn<p{2~n) for n = 1, 2, ... ■
2.11.
Pr o p o s it io n.The proper inclusion V*
c= F 0 [a , b ]holds.
P ro o f. Setting
к - 1 к
un = 2~k for Y j mm < n ^ Z w7"’ k , n = 1 ,2 ,...,
m = 1 m = 1
in view of Lemma 1.2 there exists x e R 0[a, b] such that var^x) = ]Tn<p(u„) for every ^-function (p. Given (реФ^, by Lemma 2.10 there exist d, r > 0 such that (p{2' n) ^ dr~n for n = 1 ,2 ,... Hence
var^(x) = J>(w„) = Y kk(P(2~k) > =
oo.n к к
Thus, by ([12], 1.01), х ф О ^ У * . m
3. The intersection of all classes V9. We denote by P [a, b] the linear space of real functions defined on [a, b] such that there exist a partition n: a = t1 < ... < tn = b of [a, b] and real numbers cl , ..., c2„~\ such that
*(£,) = ct for i = 1, ..., n and x(t) = c„ + i for £ e (£;, ti+1), i = 1, ..., n — 1. We set P 0[fl> b] = { x eP [a , b]: x(a) = 0}. We denote by F P 0[a, b] the linear space of regulated functions x: [a, b] -►/? which vanish at a and whose sets of values are finite. Finally, we denote by МУ*{a, b] (or shortly JO ? if [a, b] is fixed) the family of all strictly monotonical sequences in [a, b].
3.1.
Le m m a.I f x:
[ a , b ]vanishes at a then the following statements are equivalent: ■
(i) x e P0 [a, b];
(ii)
x e F R 0[a,
b ] ;(iii) /or erery sequence (tn) e J U f there exists a positive integer N such that x(tn+1) = x(tn) for n ^ N.
P ro o f. Since the implications (i)=>(ii) and (ii) =>(iii) are obvious, we shall
prove only the implication (iii)=>(i). First, observe that for a function
Remarks on functions o f bounded (p-variation 157
У' [a, [I] ->R if уфР[a, jS] and ye(a, j8) then either y|[a>y]^ P [a , y] or у|[У)/п£
P[y, /?]. Given x<£P0[a, b] such that x(a) = 0, we hence deduce that there exists a sequence ([a„, /?„]) of subintervals of [a, b] such that /?„ — а„->0, [aB + i, Æ„ + 1] c [aB, a n d x |[am/8n]£ P [ a B, /У for n = 1, 2, ...T hus there exist se [a , b] and a decreasing sequence (AB) of positive numbers tending to 0 such that either
(A) X\[S>s+A„]^P[s, S + 2 J for и = 1, 2, ..., or
(B) l[s — An,s] Ф P for n = 1 ,2 ,...
Assume that (A) holds. Then s < b. Let tx = b. Then there exists £2e(s, У such that x(£2) Ф x(tx). Otherwise we would have
(xls) for t = s,
1л II
for £e(s, £],
which contradicts (A). If (£*)"= x (n ^ 2) is a decreasing sequence such that tx — b, ti+le(s, £г) and x(ti+1) ф x(£f) for i= 1, n —1, then by (A) there exists tn + 1e(s, £„) such that x(tn + 1) Ф x(tB).
In this way we construct a sequence ( such that x(£n + 1) Ф x(tn) for all n. Thus, x does not fulfil (iii). In case (B) the proof is similar. ■
3.2. P
r o p o s it io n. The following identities hold :
n
v ° =П
К =П
K= FKoD». Ч-
<реФ <реФ (реФ
P ro o f. For every (^-function (p and every x e F R 0[a, b] the set of values of the function /^>x: Д [а, Ь]->Л, rci—►var^x; n), is finite, hence FjR0[a, b]
С П<РеФ V<p-
Conversely, suppose that there exists x e ( f ] (pe0V(p)\FRo[a, b~\. Then by Lemma 3.1 there exists a sequence {t„)eJf6F such that x(£„ + 1) Ф x(£„) for all n.
Since by Proposition 2.4, x is regulated, it follows that all elements of the set {c e R : 3n = 1 ,2 ,... |x(£„ + 1) —x(£„)| = c} may be arranged in a decreasing sequence (b„) convergent to 0. Setting
<p(0) = 0, (p{bn) = n 1 for n = 1, 2, ..., (p(u) = 1 — bx + u for и > bx,
<?(“) = (P- ~ - ^ bn+- - ( u -b n + l) + (p{bn + 1) for мб[Ьи + 1, b j , n = 1,2, ..., b „ -b n + 1
we obtain a ç?-function (p such that
var^x) S* $>(M £„ + i ) - x ( y |) ^ = 00 •
Thus, and we get a contradiction.
158 F. P r u s - W i s n i o w s k i
Now, it is enough to show that V* cz
(~)<р€ф У р -Given a ro-function (p, it is easy to see that the function cp defined by <p(u) = cpQu) for и ^ 0 is a ^-function and <p< ф (cf. the proof of Lemma 2.5). Thus by ([4], 4.4) we have
Vf P U К -
3.3. L
e m m a. For every cp-function cp there exists феФ_ such that ф(и) ^ (p(u) for и ^ 1.
P roof. Let cp be an arbitrary ^-function and let x be a strictly increasing (^-function such that cp(u) ^ yfu) for all и ^ 0. Setting un = yf2~n+l) for n = 1,2, ..., we get a decreasing sequence of positive numbers convergent to 0.
We define q : [0, oo)-»[0, oo) by ^(0) = 0, ^(tq) = x ~ 4 u2) an<3
q
{
u„) = m in i— —- — ( р Ы п - ф ^ у Чип+А for n — 2, 3, ...
K - i- M n + i 2" J
Finally, we define
q(
u) to be linear on [w2, 00 ) and on each [un+l, wj. As in the proof of 2.4 it is easy to check that деФ + and
q
{
u) ^
q(u„)^ X_1( m «+ i ) ^ Z_1(w) for ме[ми+1, u j , n = 1, 2, ...
Therefore, setting ф = £-1, we have феФ_ and <р(и) ^ /(n) ^ i/фО for w ^ 1.
3.4. C orollary . 77 ip following identities hold:
f t K °= f t K = f t V* = FR0[a, b-].
<реФ - <реФ - (реФ -
P ro o f. By Preposition 3.2 we have
f k 0[ « . 4 c f t *?<= n П v*-
<реФ - (реФ - <реФ -
Therefore, it is enough to prove that Р)<реФ- У} c FR0[a, bj. Since Ф_ а ФЛ2, we see from Lemma 3.3 and ([3], Theorem 1) that for a (^-function (p there exists феФ_ such that Vf = c Vv. Thus, by Proposition 3.2, (~)<реф_ Vf с
f]c p ^ -v< p = FR0[ a , b f ш
3.5. P roposition . For every sequence (cpn) of (p-functions the proper inclusion FR0[a, b~\ Ç\nVf„ holds.
P ro o f. Given a sequence (cpn) of (^-functions, we set
un = sup{w > 0: 2~n ^ max{(pi(nu): i = 1, ..., n}} for n = 1, 2 ,...
Since un -*■ 0, it follows by Lemma 1.2 that there exists x e R0 [a, b] such that for every ^-function cp and every 2 ^ 0 , var^/x) = 2£„<p(2wJ and such that x([a, b~\) = {0} u {un: n = 1 ,2 ,...} . Thus, x $ F R 0[a, b]. Finally, for every pair k, i of positive integers if к ^ i then
к — 1
00к — 1
00varVJ(kx) = 2 X (Pi{kun) + 2 £ ср^киф ^ 2 £ Ф*(К) + 2 £ cpfnu,,)
Remarks on functions of bounded (p-variation 159
к
— l oo^
2 Y , ( P i { k u ) f - 2 Y_,2 - ” < oo.
n = 1 n = k
This implies that xeP)„J^,°n. ■ 4. An equality of Lesniewicz type
4.1. P roposition .
F o r a r e g u l a t e d f u n c t i o n x :[a, b] -►/?
a n d a ( p - f u n c t i o n (p t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :(i) x
i s o f h o u n d e d ( p - v a r i a t i o n o n[a, b];
(ii)
f o r e v e r yb]
£</>(|x(t;+i)-x(f(.)|) < oo;
i
(iii)
f o r e v e r y t e l a ,b]
lim var (x; [t — e, f] n [a, b]) = 0 = lim var (x; (t, f + e] n [a, b]);
£-0 + E~>0 +
(iv)
f o r e v e r y t e l a ,b]
lim var„(x; [ t - e , t ) n [ a , b]) = <p(|x(t-)-x(t)|),
£-►0 +
lim varJx; [t, t + e] n la, b]) = <p(|x(t + ) —x(t)|),
£ - 0 +
(v)
f o r e v e r y t e l a ,b]
lim var (x; [t — e, f)n [u , b])
£->0
+
= max{<p(|x(t + ) - x ( t - ) |) , p(|x(t + )-x (t)|) + ç>(|x(t)-x(f-)|)}.
P roof. (i)=>(ii). Suppose that (ii) does not hold, i.e. we have oo for a sequence
( t ^ e J / f F .For
n —1, 2, ... we define a partition %n = (а")"=/e Я [«, b] as follows:
s" = tf for
i =1 , . . . ,
n +1 whenever (tj is increasing,
s i
=
t n - i+ 2 for
i= 1, ..., w+1 whenever (q) is decreasing.
We have
var<p(x; la, b]) ^ sup{var„(x; яи): n = 1 ,2 ,...}
n
= sup{ £ <p(|x(t; + 1) —x(t,-)|): n = 1, 2, ...} = 00 .
i = 1
Thus, x is not of bounded ^-variation on [a, b].
(ii) => (iii). Suppose that (iii) does not hold. Then there exists te la , b] such
that we have either the inequality lim£^ 0 + var^x; [f — e, t) n [a, b]) > 0 or
limE_+0+ var(p(x; (t, t + e] n [a, b]) > 0. We will consider only the first case. Set
lim£^ 0+ var(p(x; [t — e, t) n [a, b]) =
2 g> 0. Then t > a. Since var(p(x; [t — e,
t)n la, b]) is a nonincreasing function of e, it follows that for every ee(0, t —a]
160
F. P r u s - W i s n i o w s k ithere exists a partition nE = e l l [ t — e, t) such that t\ = t — E and var^x; nE) > g. Setting = t — a and e„+1 = t — ts^(En) for n — 1,2, ... and arranging all elements of the set (tf*: n = 1 ,2 ,..; i = 1, ..., m(e„) —1} in increasing order, we get a sequence (гп) е ^ У [ а , b] such that
m(en) — 1
2 > ( l* ( ^ + i) - * ( y ) = Z Z <p(\x(tin+ i)-x (ttn)\)
n n i = 1
= Z var«>(*; O ^ Z # = ° ° •
n n
Thus, (ii) does not hold.
(iii)=>(iv). Assume that x satisfies (iii). Given a point te{a, b) and ee(0, min (t — a, b — f}), we have by Lemma 1.1
<p(|x(t-e)-x(t)|) ^ var^x; [t - e , t]) < var^x; [t - г , t)) + ç>(oscx[ t - e , t]).
Since x is regulated and <p is continuous, it follows by our assumption that ç>(|x(r — ) — x(t)|) = lim <p(|x(t —e) —x(t)|) ^ lim var (x; [f — e, t])
£->0 + £->0 +
^ lim ^ ( o s c ^ t- e , t]) = <p(|x(t — ) —x(t)l)-
£-♦0 + Similarly it may be proved that
lim var (x; [b — e, b]) = <p(|x(b — ) — x(b)|)
£-►0 + and that for every te [a , b)
lim v arJx ; [t , t + e]) = <p(|x(t) —x(t + )|).
£-0 +
(iii)=>(v). Suppose that (iii) holds. Given te(a, b) and ee(0, min {t — a, b — t}), for a partition n = ■{ti)"=1 e l l [ t — e, t + e] we have:
if t$ n then by Lemma 1.1
var^x; n) < var^x; [ t - e , tj) + (p(oscx[ t-B , t + e]) + var„(x; (t, t + e]);
if t — tj for some j = 1, ..., n then
var<p(x; n) = var^x; {t$=1) + var^x;
< var^(x; [ t - e , t]) + var<p(x; [t, t + e]), and by Lemma 1.1 we get
var^(x; n) ^ var^fx; [ t - e , tj) + (p(oscx[ t - s , t]) + var(/,(x; (t, t + 8~\) + (p(oscx[t, t + e]).
Therefore
Remarks on functions of bounded (p-variation 161
mzLx{(p(\x(t — e) — x{t + e)|), <p(\x(t — e) — x(t)|) + <p(\x(t) — x{t + e)|)}
< var^(x; [ t - e , t + e ])^ var^fx; [ f - e , f)) + var„(x; (t, t + e]) + max{(p(oscx[t —£, t + e]), <p(oscx[t — e, t]) + cp(oscx[t, t + £])}.
Taking limits as £-»0+, we obtain by our assumption lim var^x; [t — e, t + e])
£“► 0 +
= max{<p(|x(£ + ) —x(£ —)|), <p(|x(£ + ) x(t)|) + <jo(|x(t ) x(t)|)}.
((iv) or (v))=>(i). If (iv) or (v) holds then for every te[a, b] there exists an sf > 0 such that var<p(x; [t — £t, t + e j n [u, b]) < oo. From the cover {(£ —£f, £ + £f) n [a, b]: ts [ a , b]} we choose a finite subcover {(tf — eti, H - .) n [a, b] : i = l , . . . , m } . Then there exists a partition я = (sf)î"=Y such that s1 = a, sm+ 1 = b and for every i = 1, ..., m there exists j = 1, ..., m such that [sf, S|+1] cz(tj—8tJ, tj + Efj) n [a, b]. By monotonicity of (^-variation with re
spect to set inclusion we have var^(x; [s£, si + 1]) < oo for i — 1 , m. Since x is bounded on [a, b], it follows by Lemma 1.1 that
m
var„(x; [a, b]) ^ £ var„(*; 1Л> si+1]) + (m -l)^(o scx[a, b]) < oo.
i = 1
This finishes the proof of the implication ((iv) or (v))=>(i). Thus the proof of 4.1 is complete. ■
For every ^-function q> and every function x of bounded cp- variation on [a, b],
( + ) var^(x; [a, b]) ^ 8ир{£>(|х(£; + 1)-х (£ ;)|): {QeJffF}.
i
For the function
fO for t = a or t = b,
^ (1 for te(a, b), and for every ^-function cp we have
varg)(x; [a, b]) = 2q>(l) > q>( 1) = s u p ^ ( |x : ( t £+ 1 ) —^c(t£)|): {tJeJtS?}.
i
It is easy to see that in ( + ) equality holds iff x is continuous at a or at b.
For a ^-function (p and a sequence we set
U (*,•)) = { x e X : (I x i U + J - x i t J D f L ^ Q , /*((Y) = { x e X : (\x(ti+1) - x ( t i)\)^1el*}i Ç((Y) = { x e X : (|x(ti+1)-x(Y|)?°=ie^}, 4.2. C orollary . For an arbitrary (p-function q>,
П — Commentationes Math. 30.1
162
F. P r u s - W i s n i o w s k iKp= П U(*.-)) and v<? = П Wh))-
(i>i)e.M£f (ц)е.Ж У
4.3. P
r o p o s i t i o n. For an arbitrary cp-function (p,
k
* = n №«))•
P roof. It is obvious that V* c f ] (ti)ej(y Conversely, let x e R 0[a, bJ\V*. Then by Proposition 4.1 for every positive integer n there exists a sequence (t?),® i fr] such that £;<р(«-1 |x(t"+1) —x(t")|) =
g o.Without loss of generality we may assume that those sequences are either all increasing or all decreasing. We will consider only the first case. Let gn = lim^"
for n = 1 ,2 ,... and let ge[a, b] be a cluster point of (gn). Then either (i) for every e > 0 and « = 1 ,2 ,.* . there exists a decreasing sequence (*",e),“ i e J t ^ d g - s , g) n [a, b]) such that Wt?+i)-x(*",e)l) = 00 or
(ii) for every e > 0 and « = 1 , 2 , . . . there exists an increasing sequence + b]) such that £ ip (« " 1|x(t?:he1)-x (t7 ’e)l) = oo.
We will consider only case (i). If (i) holds then g > a. For se(0, g — a] and an integer « > 0 let (t",E)fLi be an increasing sequence from JiSf\_g — e, g) such that
t ni ’e = g — £and ^ £ф(«_ 1 x) — x(t",£)|) = oo. Writing
k
m(n, e) = min{/ce{1, 2, ...}: £ ^(n-1 Ml?+£i)-x (l? ’£)l) ^ 1},
i= 1
we set £x = g —a and s„ + 1 = g — t m(n<£n) for « = 1 ,2 ,... Arranging all points t",E ( « = 1 , 2 , . . . ; i = 1, ..., m(n, sn)) in increasing order, we get a sequence ( t^ E J é ^ \a , b] such that for every positive integer к we have
m(n,en)
£>(fc~1|x(ti + 1) - x ( t i)|) = Z Z <p(fc~1|x (# ft)-x (tf,Cn)|)
i n i = 1
oo m(n,e„) oo
^ Z Z ф ( и _1И № 1 )- х (^")1) ^ Z 1 = °°-
n = к i= 1 n=k
Thus, x£/*((t*)), and this proves that **(&)) c ■ 4.4. P
r o p o s i t i o n. Lei q> be a cp-function. Then
K *
.= n *? = n k?-
ф <<p ф о cpP ro o f. We have l* = ([5], p. 111). Thus, by Corollary 4.2 and Proposition 4.3 we obtain
v % = n ш ) = n п'Ж))=л л ф- m ) = n c
4<р фЧ(р (ti)Ejf/£f ф<(р
Since if ф (p then ф<ир, the second equality follows by ([4], 4.4) from the first
one. ■
Remarks on functions o f bounded cp-variation 163
If \j/<3(p and ф is not equivalent to <p, then we will write ф<з<нр. Observe that tp-<up iff (реФА2. Hence if ф<нр and (рфФд2 then фс<ир. Moreover, if ф (p then ф<з<ир.
4.5. P roposition . For every sequence (фп) of (p-functions such that фп-<з<ир for all n the proper inclusion Vf cz f ) n Vfn holds.
P roof. Observe that tДо<]<р iff the proper inclusion /* <= /$ holds (cf. [9], 2.5, 2.51 and 2.41). Thus if ф<з<ир then for every у > 0 there exists a decreasing sequence (un) convergent to 0 such that u1 < у and Z „ф{сип) < oo and = 00 f°r every c > 0. Given a sequence (фп) of ^-functions such that фп< кир for all n, we set x„(u) = max{i/q(u): i = 1, ..., n] for и ^ 0 and n= 1 ,2 ,... Then all %n are ^-functions and
(+ ) for * = U 2, •••;
( + + ) Xn(u)^Xn + i ( m ) for и > 0, n = 1, 2, ...;
(+ + + ) П F*°„ = П
K -* « n
By ( + ) for every n = 1 ,2 ,... there exists a decreasing sequence (n£)®= i of positive numbers convergent to 0 with u" < n_1 and there exists an integer mn > 0 such that
m n m n
(+ + +.+ ) Z
X n ( n u k)< 2~ " and Z <jp(n-1 M2)^ 1.
fc=i fc=i
Arranging all numbers иЦп = 1 ,2 ,...; к = 1, ..., m„) in nonincreasing order, we obtain a sequence (ufe) of positive numbers convergent to 0 such that for every pair m, / of positive integers, setting r = max{m, /}, we have by ( + + ) and (+ + + +)
m n r — 1 m„
oo
m nZ Zm(4) = E I Xm (fofc) < Z Z Х«(*“к) + Z Z &.(”“*) < 00 »
к n k = 1 n ~ 1 к — 1 л — г /с — 1
m n
oo
Z<p(/-4) = Z Z Z Z = со.
fc n k = 1 n = r /с = 1
Thus, by Lemma 1.2 and ( + + + ) there exists х е Я 0[а, b] such that x e P)„ Vfn and хф Vf. ш
4.6. C orollary . I f cp ф ФЛ2 then for every sequence (фп) of (p-functions such that фп<мр for all n the proper inclusion Vf c f ] nVfn holds.
For every sequence (фп) of (p-functions such that фп cp for all n the proper inclusion V* c= P|n Vfn holds.
From Corollary 4.6 it follows that ф <p implies that the proper inclusion
F* c F j holds. Thus {J^^yV f a Vf. But for modular classes of functions of
164
F. P r u s - W i s n i o w s k ibounded (^-variation the equality analogous to equality (I) from [7] does not hold in general. Taking the (^-function ф(и) = и, we see that every nonconstant linear function x e X belongs to If ф cp, then by ([3], Lemma 4) no nonconstant linear function x e X belongs to F*. Thus, Ф V,f for the
^-function ф(и) — и.
5. (^-variation and well-regulated functions. A function x:
[ a ,b] -> R is said to have internal saltus at a point t0 of discontinuity if
lim infx(t) < x(t0) < lim supx(t).
t - * t o t - * t o
We will call a regulated function x: [a, b] -> R well-regulated if x is continuous at the endpoints of [я, b] and has internal saltus at each point of discontinuity.
5.1. P roposition . Let (p be a superadditive (p-function and let x be a well-regulated function defined on [ a , b]. Then if G is a dense subset of [ a , b]
then var^x; [a, b]) = var<p(x; G).
P ro o f. Since the «^-variation is monotonical with respect to set inclusion, it is enough to show that varç)(x; [a, b]) ^ var<p(x; G).
If var<p(x; [a, b]) = (p(\x(a) — x(b)|) then taking a decreasing sequence (t„) in G convergent to a such that tx < b, and taking an increasing sequence (sn) in G convergent to b such that > t lt we define for n = 1, 2, ... the partition nn = {r")f= ! eIJG as follows: r" =± tn, r\ = sn. Since x is continuous at a and at b, by the continuity of cp we have
var„(x;
7T „) = =<p(|x(sj-x(t)|)-><jf>(|x(b)-x(a)|).
Thus,
varv(x; G) ^ sup{var„(x; n„): n = 1 ,2 ,...}
> <p(Mb)-x(a)|) = var<p(x; [a, b]).
If var^x; [a, b]) > (p(\x{b) — x(a)\) then
var<p(x; [a, b]) = sup{var(p(x; n): 7 üg Л [а, b] and cardrc >3}.
Take £ > 0 and a partition n = (?,)"= 1еП{а, b] with n ^ 3 and choose Ô > 0 so that
( + ) (p(u + 2S) ^ (p(u) + e/n for
m g[ 0 ,
oscx[
m, b ] ] .
Because <p is superadditive we may assume that
(x(ti+2) - x ( t i+1))(x{ti+i)- x { tù ) < 0 for i = 1 ,..., и —2.
We will consider only the case x(t2) —x(£t) < 0. The remaining case is
considered similarly. Let 2y = min{|£i+1 — tf. i = 1 , . . . , n — 1}. Since G is
dense in [a, b] and x is well-regulated, for и, there exists
Remarks on functions o f bounded (p-variation 165
— y, l;+ y )n G such that
x(st) < x(t() + <5 for even i, x(st) > x(ti) — 3 for odd i.
Then
2^ + |x(si+1) - x ( s 1)| ^ |x(fi+1) —x(f{)| for i = 1, n - 1 . Thus, by ( + )
n — 1 n — 1
8+ X <p(l^(Si + i)-x(Si)|)> X <p(2^ + |x(sf + 1) - x ( s i)I)
i = 1 i = 1
n - 1
^ X <P(|x(ti+i)-*(*i)l) = var„(x; я).
i = 1
Since n' = (s,-)"= i is a partition of G, it follows that var„(x; G )^ v a r<p(x; я') ^ var„(x; я )-е . Therefore, var^x; G) ^ var^x; [u, b]). ■
If a ^-function <p is not superadditive then there exist ul , u2 > 0 such that q>(u1 + u2) < <p(u1) + (p(u2), and therefore, for an arbitrary ce (a, b) and for the well-regulated function
{ 0 for te[a, c), ux for t = c, ul +u2 for t e(c, b]
we have
var^(x; [a, b])
=+ >
ф ^ + и ^= var^x; [a,
c) v j(c,b]).
5.2. C orollary . / / a (p-function (p is superadditive and x: [a, b ]-*R is well-regulated then the (p-variation of x on [a, b] is independent of the values of x at its points of discontinuity.
5.3. C orollary . Let q> be a superadditive (p-function. I f two well-regulated functions x ,y : [a, b]->R coincide at their points of continuity then they have the same (p-variation on [a, b].
5.4. C orollary . Let q> be a superadditive (p-function and let x: [a, b]-»R be a regulated function. I f y : [a, b~]-+R is a well-regulated function such that x = y at the points of continuity of x then var<p(x; [a, b]) ^ var^y; [a,‘ b]).
P ro o f. Denoting by Cx the set of points of continuity of x, we have y|Cx = x|Cx and Cx is dense in [a, b]. Thus, by Proposition 5.1 and by monotonicity of the ф-variation with respect to set inclusion we obtain
var„(y; [a, b]) = var„(y; Cx) = var<p(x; Cx) ^ v a rjx ; [a, b]). ■
This corollary with 5.2 implies that if x is of bounded ф-variation on
166
F. P r u s - W i s n i o w s k i[a, b] then among all functions continuous at the points of continuity of x the well-regulated ones have minimal tp-variation on [a, b].
References
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INSTYTUT MATEMATYKI, UNIWERSYTET SZCZECINSKI INSTITUTE O F MATHEMATICS, UNIVERSITY O F SZCZECIN WIELKOPOLSKA 15, 70-451 SZCZECIN, POLAND