On functions of bounded essential variation

ROCZNIKI POLSK.IEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

A. G.

and B. K.

(Kalyani, India)

On functions of bounded essential variation

1. Introduction and definitions. The notion of essential variation of functions was introduced in [5] to obtain some function theoretic properties in an abstract space. This idea was used in [6] to obtain some properties of derivatives and integrals relative to such a function in addition to a suitable generation of signed measure induced by such a functional or its ancillary. In this paper our purpose is to prove some fundamental properties of functions of bounded essential variation. We also construct certain pseudometric space of the class of such functions in the light of papers [1], [4]. Property A* as used in Theorem 3.1 first appeared in [3]. The idea of using polygonal functions to the theory of functions of bounded variation appeared first in [2] and was used in [1] and subsequently in [4] to obtain separability of the spaces of functions of bounded variation as constructed in [1] and [4] in two different contexts.

We quote below some lemmas and definitions some of which are new and others are borrowed from [5].

De f i n i t i o n

1.1 ([5]; §2). Let f { x ) be defined in the closed interval [a, b] and let E be any subset of [a, b] with Lebesgue measure, mE = b — a.

Consider a subdivision D: x0 < < ... < x„ of [a, b] with x,e E and define V [ f \ £] = SUP L l/(*«)-/(*;-i)l-

d , = 1

The essential variation of the function / in the interval [a, b] is defined by inf {V\_f ; £]: E c [a, b], mE = b - a ] and is denoted by V* [ / ; a, b]. If V* I f l a, b~\ < + oo, we say that / is of bounded essential variation in [a, b]

and we write / e BV* [a, b].

Lemma

1.1 (cf. [5]; Theorem 2.2). For every f ( x ) defined in \_a, b] there is a subset A of [a, b] with mA = b — a such that V [ f ; Â ] = V* [ / ; a, b].

In future we shall use such a set A without further explicit reference.

No t e 1.1. Without any loss of generality it may be assumed that [a, b~\\A is everywhere dense in [a, b], because if В cz A, mB = mA, then

b-] = V l f ; B l

220 A. G. D a s and B. K. La hi ri

L

1.2. I f /

BV* la, b], then lim f(Vi), v{e A , a ^ x < b and

Dj -*x +

lim f(Vi), V

A exist. Further f is continuous for

A with respect to A

iч ->b —

except at most a countable set of points.

Definition

1.2 (cf. [5]; Theorem 2.2). Lei /

BV* [a, b]. Define f *(x) in [a, b] as follows:

f * ( x ) = lim f(Vi), V

A , a ^ x <b,

t’i ~ * X +

/*(&) = lim f(Vi),

A,

vi -*b —

and call f *( x) the reduced function of / in la, b].

L

1.3 ([5]; Theorem 2.2). I f f

BV* [a,ft], then the reduced junction f * is right-hand side continuous for a ^ x < b and left-hand side continuous for x = b. Further f * (x) equals f almost everywhere in [a, b].

L

1.4 ([5]; § 2, Corollary 1). I f / e B F * [ a , b], gGBV*la, b] and f ( x ) = g ( x ) almost everywhere, then f * (x) = g* (x) for a < x

b and V* I f ; a, b] = V*lg; a, b] = F [ / * ; a, b] = Vlg*; a, b].

From the definition of reduced function and Lemma 1.4 we obtain L

1.5. I f f G B V * l a , b] and f is right continuous in [a, b) and left continuous at b, then V* I f ; a, b] = F [ / ; a, b].

2. Some results on bounded essential variation. If / е й К [ а , b], then clearly / e BV* [a, b]. But the converse is not true as may be seen by considering the well-known Dirichlet’s function ф(х) which equals 1 for rational x and 0 for irrational x in [0, 1].

We first prove the following lemma.

L

2.1. For every junction F(

)

BVl a, b] there exists a junction /(x )e B F * [a, b ] - B F [ a , b] such that /( x ) equals F(x) except for a countable set of points and V* [ / ; a, b] = V* [F; a, b].

Proof. Let F(

)

B V la , b] and let ]Tr„ be a divergent series of positive terms with as n-> oo. Let {u„} be a strictly increasing sequence contained in an infinite subset G of la, b] with mG = 0. Define /( x ) in la, b]

as follows:

f (x) = F (x) + vn for x = u2n (n = 1, 2, ...)

— F(x) elsewhere.

Let E a la, bJ\G and mE = b —a. Consider any subdivision D: x0 < x x

< ... < xm of la, b] with

(

E. Then

m m

L = I If ( x ,) - F (*,-,)! « K[F; a, 6].

/= 1 i = l

This gives V* I f ; a, b] ^ F [ / ; £ ] ^ F [F ; a, b] and so f GBV * la, b].

N ext, consider a su b d ivision a ^ ux < u2 < . . . < u2p < b o f [а , b]. T hen

\f (us) - f (a) \ + £ | / ( « i) - / ( « 1- 1)l + l / ( b ) - / ( » 2p)l

i =

^ Z l / ( W2f)-/(W2i-l)l = Z i^'(W2i) + ^-^(W2i-l)l

i = 1

1

5

V , - t

i —

1 i = l

i= 1

Since F e B V[a, b] and Z X is divergent, it follow s that / ф В У\_а, b]. T he p ro o f is n ow co m p lete follow in g the definition o f f (x) and L em m a 1.4.

U sin g D efin itio n 1.1 and L em m a 1.1 w e obtain

Lemma 2.2. I f a (x ) and (i(x) are o fbounded essential variation in [a , b], then o t± p is also o f bounded essential variation in [a , b] and V* [ot± f i; a, b] ^ V* [a ; a, b] + F *[j3; a, b].

Corollary 2.1. I f a, f are o fbounded essential variation in [a , b], then F* [ a ± 0 ; a, b] ^ |F * [a ; a, b ] - F * [/?; a, b]|.

Theorem 2.1. I f f (x) and f„ (x), n = 1, 2, . . . , are defined on [a , b] and {/„ (x )} converges to f (x) alm ost everyw here in [a , b], then

(2.1) lim inf F * [/„ ; a, b] ^ V* [ / ; a, b ].

л -»ao

P r o o f . Let G be a subset o f [a , b] such that {/„ (* )} converges to f (x)

at each p oin t o f G. W e first su p p ose that F* [ / ; a, b] < + o o . C onsider a subset JE o f G w ith mE = b—a and F [ / ; £ ] < + o o . Let £ > 0 be arbitrary.

There exists a su b d ivision D: u0 < u t < . . . < um o f [a , b] w ith u , e £ such that:

I K [/;£ ]-e .

i= 1

Since /„ ( ц ) - > / ( ц ) as n -> oo for each i (i = 0, 1, . . . , m), there is an n0 such that for n ^ n0 w e have |/„ (n t) —/ (w,)| < fi/2m, i = 0, 1, . . . , m. F or n ^ n0 , w e have |Л ( Ц - ) - / п(м ,-1 )1 + £ Х > I / ( « / ) - / ( « . - i ) i for г = 1 , 2 , . . . , m . T hen for n^ n0

m m

Z

Z

i =

i=l

So, F [ / „ ; £ ] + £ > F [ / ; JE]—£ ^ V* [ / ; a, b] — s. Since mE = b —a, £ c: G is

222 A. G. D as and B. K. Lahiri

arbitrary and addition of more points on E does not decrease the sum, it follows that

(

.

) V* l fn ;a ,b l + e > F * [ / ; a , b ] - £.

Inequality (

.

) is true for each n ^ n0 and hence lim inf V* [/„; a, b] ^ V* [ / ; a, b].

Л

-*00

I f

у* [ / ; a, b] = + oo, then V [ / ; £ ] = + oo for E a [a, b] with mE

= b — a. Let E 1 be a fixed subset of [a, b] with mFj — b — a and c G.

Then for e > 0 arbitrary, there exists a subdivision Dx \ u0 < u1 < ... < up of [a, b] with uie E 1 such that

X l/(Mi)-/(M,--i)l >

/e.

i =

The procedure as above shows that V* [/„; a, b~] + £ > 1/e for n ^ n ^ e ) . Hence lim inf F* [/„; a, b] = + oo. This completes the proof.

Л * GO

N o te 2.1. It may be noted that everywhere convergence of the sequence {/„(x)}

/(* ) or even the uniform convergence is not sufficient to ensure the equality in (2.1). For example, consider f n(x) = ^sin nx, 0 ^ x ^ я. Then {f„(x)} converges uniformly to / (x) = 0 in [0, я]. By Lemma 1.5, F* [/«; л] = F [/„; 0, я] = 2 for each n and V* [ / ; 0, я] = 0. So,

lim V* [/„; 0, я] = 2 Ф V* [ / ; 0, я].

л-*оо

The equality in (2.1) will be established in the next section under certain additional hypotheses.

3. Convergence in bounded essential variation. Let g0(x) and g„{x), n =

,

, ... be defined on [a, b] and let (gn{x)} converge to g0{x) at each point of a subset G of [a, b] with mG — b — a. By Lemma 1.1, there exist An c G with mA„ = b — a such that V[g„', An~] = F* [gn; a, b], n = 0, 1 ,...

Let A0 = 0 An so that mA° = b — a and V[gn\ А0] = F * \_gn; a, b] for each

n= 0

n >

.

Property

A*. The sequence (#„(*)} is said to possess Property A* on

[a, b] if a subdivision D0:

< £i < ••• < ^ °f [a, b] with and a

positive integer in exists such that \g„{x') — g„(x")| ^ \ут(х') — 9т(х”)\ whenever

n ^ m and x', x" belong to the same subinterval

^

^ /

+

,

C-i = a, ^ + i = b.

Let A be the collection of all subdivisions D\ x 0 < x x < ... < xp of [a, b] with ххеА°. If (p{x) be any function defined on [a, b], we write

(<P* D) = X И * ;) -<?(*;-i)l- i=

We present two lemmas without proof.

L

3.1. lim (g„, D) = {g0, D) for every De A.

n

L

3.2. I f V* \_g„; a, b] ^ К for all n, where К is a finite number, then V*[g0; a, b] K.

L

3.3. I f the sequence [gn(x)} possesses Property A* on [я, b] and if V*l9n\a, b] > K for all n, К being fixed, then a De A exists such that (g„, D)> К for all n.

P ro o f. Let DxeA where Dx => D0, D0 being the subdivision in relation to Property A*. It is easily seen that (gn, Dx) ^ (gm, Dx) when n ^ m. Since V* Q/, ; a, b] > К for each i, 1 ^ i ^ m, an element D2 of A exists such that (#, ; D2) > К for each /, 1 ^ i ^ m.

If D = Dx u D2, then De A and (gn, D)> К for all n. This proves the lemma.

L

3.4. If {#„(*)} and all its subsequences possess Property A* on [a, b] and if V* [g0 ; a, b] < К, К being fixed, then V* [<?„; a, b] ^ К for all n except possibly for a finite number.

P ro o f. If possible, suppose that the lemma is false. Then there exists a sequence of positive integers {«,] with n,--> oo such that V*[g„.; a, b] > K.

Using Lemma 3.3 and then Lemma 3.1 we obtain (g0, D) ^ К for some De A. So V*[g0; a, b] ^ K. The contradiction proves the lemma.

T

3.1. I f \g„(x)} and all its subsequences possess Property A* on [a, b] and V* [g„; a, b] is finite for each n, then

lim V* [gn; a, b] = V* [g0; a, b].

n

P ro o f. We first suppose that V*[g0; a, b] < +oo. Then there exists a positive number К such that V*[g0; a, b] < K. By Lemma 3.4, there exists an integer n0 such that V* [g„; a, b] ^ К for n ^ n0.

Let L = lim sup V* [gn; a, b] and / = lim inf V* [gn; a, b]. There exists a sequence {«,} of positive integers such that V*[gn.; a, b]-^ L, as i-> oo.

If e > 0 is arbitrary, an integer i0 exists such that (3.1) L - e < V* [<?„.; a, b] < L+e when i ^ i0.

So, by Lemma 3.2,

(3.2) V*[g0; a, b] ^ L + e.

224 A. G. D as and B. K. Lahiri

Utilizing Lemma 3.3 and then Lemma 3.2, we obtain, from the first inequality in (3.1), (g0, D) ^ L —e for some De A. This gives

(3.3) V*[g0; a,

] = V[g0; Л0] ^ L - e .

As £ > 0 is arbitrary, combining (3.2) and (3.3) it follows that V* Dio ; a> 4 = L. Similarly, we obtain V* [,g0 ; a, b] = /. Hence lim K* lgn; a, b] = V* [g0; a, b].

И“>оО

If ^ * [

o; a, b] = + oo, then there cannot exist any subsequence {nf of positive integers for which the sequence {V*[g„; a, b]} is bounded, because in that case V* [g0; a, b] would be bounded, by Lemma 3.2, and this would imply that V*[g0;a, b] is finite. Hence lim V* [g„; a, b] = + oo. This

n-*oo completes the proof.

4. The space (A, d). Let X denote the set of all functions x(f) such that x e B V * [ 0 , 1]. To each pair of functions x, у in X we associate the real number d(x, y) defined by

(4.1) d{x, у) = S\x(t)-y{t)\dt + \V*(x)-V*(y)\, о

where the integral is taken in the Lebesgue sense and K*(.x) stands for V* [x; 0, 1]. The existence of the integral on the right of (4.1), is assured by the fact that corresponding to each x e X there exists an x * e i? F [

,

] such that x* equals x almost everywhere in [

,

] and F[x*;

,

] = L*[x;

,

] (see Lemmas 1.3 and 1.4). If x{t) = y(t) almost everywhere in [0, 1] and x, y e X , then the integral part as well as the variation part in (4.1) vanish separately and consequently d(x, y) — 0. But if d(x, y) = 0, then L*(x)

= V*(y) and x(f) = y(r) almost everywhere in [0, 1]. It follows therefore that d is a pseudometric for X and therefore (X, d) is a pseudometric space.

In the following results we shall use the terms closed, sphere, compact, separability, etc., with reference to the pseudometric d for X.

Theorem

4.1. The space (X, d) is separable.

To prove the theorem we require the following definition and results.

D

4.1 (cf. [2]; §1). Let /(x ) be defined on [a, b] and E c : [a, b] with mE = b — a and V\_f\ E~] = V* [ / ; a, b]. Let D: x

< x 1

< . . . < x m be any subdivision of [a, b] with xteE. We denote by B(x)

— B (x ;/, D) the function whose graph is the polygonal line joining the

points (х,-,/(хг)),

^ i ^ m, B(x) = B{x0) for a ^ x ^ x0, B(x) = B(xm) for

xm < x ^ b. J3(x) is said to be a polygonal function associated with f (x)

relative to the subdivision D.

It is clear that

m m

I .,1 = £ |B(xi)-.B(.x,._1)|

i = 1 i — 1

- V[B\ a, ft] = V*[B; a, ft].

The last equality is obtained owing to the continuity of the polygonal function B(x) (see Lemma 1.5). It then follows that

(4.2) V* [ / ;

ft] = V [ f ; £ ] > V* [fi;

ft].

Theorem

4.2 (cf. [2]; § 2). I f f e BV* [a, ft], then it is possible to choose a sequence {Bn (x)j of polygonal functions such that B„(x) converges to f (x) almost everywhere in [a, ft] and

lim V* [Bn; a, ft] = V* [ / ; a, b].

П-+

The following lemma is needed to prove Theorem 4.2.

L

4.1. If f e B V * [ a , ft] and a ^ x ( < b, then for 8> 0 arbitrary there exists <5 > 0 such that V* [ / ; x ls x] <8 for x 1 < x < x l + ô.

P ro o f. Let f * be the reduced function of / in [x1? ft]. The right-hand continuity of /* at x l shows that to each e > 0 there exists a <5 > 0 such that V[f*; x

x] < 8 whenever х г < x < Xi + ft. This implies that V* [f*; x lt x]

<e whenever x

< x < x

+t>. Since / * = / almost everywhere in [xls Xi + <$], this yields V* [ / ; x t , x] < s whenever Xi < x < X! + ft.

P r o o f of T h e o re m 4.2. Let £ о

ft], mE — b — a and K [ / ; £ ]

— L* [ / ; a, ft]. We note first that the set of points of discontinuity of / in E with respect to E is countable (see Lemma 1.2). Let {D„} be a sequence of subdivisions Dn: x„

< х„л < ... < x„>r of

ft] with xn>Ie £ , 0 ^ i ^ rn, such that D„ a D„+l, D = u Dn is everywhere dense in [a, ft] and D contains all the points of discontinuity of / in E with respect to £. Consider the sequence (£„(x)} with £„(x) = B(x; f Dn) for each n.

We show that Bn(x)~> f( x ) at each point of E\{b}. At each point of D the convergence is evident. If £\({ft} u D), then for arbitrary e > 0 we choose (> £) in D such that \f(c)—f(b)\ < ?8 and V* [ / ; £'] < ie. The first inequality is obtained owing to the continuity of / at £ with respect to

£ and the second inequality is obtained from Lemma 4.1. There exists a positive integer N such that e Dn for all n ^ N. Then for all n ^ N

\Bn(c)~f(f)\ ^ \вя( £ ) - в я( а + \ / ( 0 - № \

< v t B n; t , n + U

= V*[Bn; £, £ ']+ ie (see Lemma 1.5)

< V* I f ; f , C'] + i e (using (4.2))

226 A. G. D a s and B. K. Lahiri

As mE — b — a, it follows that Bn(x)-^> f { x ) almost everywhere in [a, b].

From Theorem 2.1 we get lim inf V* [B„; a, b] ^ V* [ / ; a, b]. Using Л

(4.2) we obtain lim sup V* [Bn; a, b] ^ V* [ f ; a, b]. This completes the

«-► 00

proof of Theorem 4.2.

P r o o f o f T h e o re m 4.1. Let Q denote the set of all polygonal functions in X with rational corners. Clearly Q is countable.

Let x( t)eX. By Theorem 4.2, it is possible to choose a sequence of polygonal functions {#„(£)} in X such that #„(£)-» x(t) almost everywhere in [0, 1] and V*(Bn)-+ V*(x). For each Bn(t) we can choose a polygonal function Pn{t) in Q such that \Bn(t) — Pn{t)\ < l/n everywhere in [0, 1] and

\V*(Bn)—V*(Pn)\ < l/n. So, the sequence {Pn(t)} converges to x(f) almost everywhere in [0, 1] and F*(P„)-+ F*(x). Therefore d{Pn, x) -* 0 as n —> oo.

This shows that x is an accumulation point of Q and hence Q is dense in X.

This completes the proof.

N o te 4.1. Whenever Theorem 4.2 is obtained, the proof of Theorem 4.1 is verbatim with the proof of Theorem 4.1 of [1]. However, for the sake of completeness, we give the proof.

Theorem

4.3. No sphere in (X, d

is compact.

P ro o f. Let у be a sphere in {X, d) with centre ct(t)eX and radius r (# 0 ). Consider the sequence {x„(£)} on [0, 1] as follows:

x„(t) = cc(t) + a„(t),

where a_(£) = —sin nut, 0 < £ < 1, \K\ ^ r/3, n = 1,2, ... Clearly {x„(£)|

n

converges uniformly to a (t) in [0, 1]. Further, by Lemma 1.5, V*(cc„) — F(a„)

= 2\K\ for each n so that

„

X. That x„e X follows from the inequality

^ V* (a) + V* (°0 (see Lemma 2.2). Also

d (x„, a) = f |x„ (f) - a (f)| dt + \V* (x„) - V* (a)|

о

^ I la«(OI dt 4- F*(a„) о

^\K\/n + 2 \ K \ ^ 3 \ K \ .

Since |X| ^ r/3, d(xn, a) ^ r for each n so that each x„ey.

If possible, let у be compact. Then there is a subsequence {x„.} of [x„}

which converges to an element x of y. Now d(x„., x)-»0 as / -*■ oo implies l

each of j|x„.(£)-x(f)|dt and | V*{*„) — V*(x)| tending to 0 as i-> oo. Uniform

о ‘ * l l

convergence of {x„.(f)} to a (t) implies lim J|x„.(£) — x{t)\dt = J|a(t) — x(t)\dt

n->0о

and so x(t) = a(f) almost everywhere in [0,1]. By Lemma 1.4, F*(x)

= V*(ct) and by Corollary 2.1, IV* (x„.) — V* (x)| = | V* (a + a„.) — V* (a)|

^ F*(a„.) = 2\K\ for each i. The contradiction proves the theorem.

Th e o r e m

4.4. The space (X , d) is not complete.

P roof. Consider the sequence {p„{t)} on [0, 1] defined by P„(t)

= -sin nnt, n = 1, 2, ... By Lemma 1.5, V*(P„) = 2 for each n and so P„eX.

n

For any two positive integers m and n

d(Pm, P„) = \Pm( t ) - P M d t + \V*(pm) - V * ( p n)\

01

sin mnt m о

sin rmt

n dt

1 1

^ — I—

. m n

So, d(pm, pn)-> 0 as m, n-> oo and consequently {/?„} is a Cauchy sequence in (X , d). If \P„) converges to a limit ft in (X, d), we should have

(4.3) lim f|A,(r)—/?(«)|Л = 0 and lim |F * (W -K » (« | = 0.

п -►x о П-+Ю

The first shows that P(t) = 0 almost everywhere in [0, 1]. If p e X , then V* (p) = 0 and so IV* {P„) — V* (P)| = 2. This contradicts the second relation of (4.3).

References

[1] C. R. A d a m s, The space o f functions o f bounded variation and certain general spaces, Trans. Amer. Math. Soc. 40 (1936), 421-438.

[2] —, and H. L ew y, On convergence in length, Duke Math. J. 1. (1935), 19-26.

[3] P. C. B ha к ta, On bounded variations, Ganita 14 (2) (1963), 59-65.

[4] —, On functions o f bounded variation relative to a set, J. Austral. Math. Soc. 13 (3) (1972), 313-322.

[5] Z. C y b e r t o w ic z and W. M a tu s z e w s k a , Functions o f bounded generalized variations, Comment. Math. 20 (1977), 29-52.

[6] D. N. S a rk he 1, On measures induced by an arbitrary function and integrals relative to a function o f bounded residual variation, Revue Roum. Math. Pures Appl. 18 (1973), 927-949.

d e p a r t m e n t o f m a t h e m a t ic s UNIVERSITY O F KALYANI WEST BENGAL. INDIA

— Prace Matemalyczne 26.2