Integration between the Lebesgue integral and the Henstock-Kurzweil integral : its relation to local convex vector spaces

INTEGRATION BETWEEN THE LEBESGUE INTEGRAL AND THE HENSTOCK-KURZWEIL INTEGRAL Its Relation to Local Convex Vector Spaces

SERIES IN REAL ANALYSIS

Vol. 1:

Lectures on the Theory of Integration R Henstock

Vol. 2:

Lanzhou Lectures on Henstock Integration Lee Peng Yee

Vol. 3:

The Theory of the Denjoy Integral & Some Applications V G Celidze &AG Dzvarseisvili translated by P S Bullen

Vol. 4:

Linear Functional Analysis WOrlicz

Vol. 5:

Generalized ODE S Schwabik

Vol. 6:

Uniqueness & Nonuniqueness Criteria in ODE R P Agarwal & V Lakshmikantham

Vol. 7:

Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces Jaroslav Kurzweil

Vol. 8:

Integration between the Lebesgue Integral and the Henstock-Kurzweil Integral: Its Relation to Local Convex Vector Spaces Jaroslav Kurzweil

Series in Real Analysis - Volume 8 INTEGRATION BETWEEN THE LEBESGUE INTEGRAL AND THE HENSTOCK-KURZWEIL INTEGRAL Its Relation to Local Convex Vector Spaces

Jaroslav Kurzweil Mathematical Institute of the Academy of Sciences of the Czech Republic

V f e World Scientific wb

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INTEGRATION BETWEEN THE LEBESGUE INTEGRAL AND THE HENSTOCK-KURZWEIL INTEGRAL Its Relation to Local Convex Vector Spaces Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

The approach to integration by Riemannian sums was rehabilitated in the fifties of the 20th century by a new interpretation of the concept of a "fine" or "6-fme" partition of the integration interval. It is well known that both the Lebesgue integration and the Henstock-Kurzweil integration can be obtained by the same method, only the integration bases are different. The concept of an integration basis y is very flexible and results in a rich class of ^-integrations. To every integration basis 3^ there corresponds the vector space Py of primitives of 3^-integrable functions (on a fixed interval / = [a, b] C K), a concept of Ey-convergent sequence of functions from Py, and ULci^y) which is the finest locally convex topology on Py such that every Ky-convergent sequence is convergent in (Py:ULc(^y))Lebesgue integration is obtained by a suitable choice of y, y = C. Then Pc is the space of absolutely continuous functions and ULC(^-C) is induced by the norm ||.F||var = vari*1. Hence (Pci^Lci^c)) is a complete space. If y — TiK, then HenstockKurzweil integration is obtained: The topology Uici^wc) is induced by the norm H-FHsup and (PHKMLC^HK)) is not complete. The problem whether (Py ,UL,c(J&y)) is complete is the central problem of this book. A theory is developed which gives an answer for a broad class of J^'s and to an extended problem which includes integrations introduced by Bongiorno and Pfeffer in 1992 and by Bongiorno in 1996.

V

PREFACE

VI

Topics connected with the Riemann approach to integration were reported and discussed in the Seminar on Differential Equations and Real Functions of the Mathematical Institute of the Academy of Sciences of the Czech Republic since the beginning of this approach. I wish to thank the participants of the seminar for their contributions and comments. I express sincere thanks to J. Jarnik and S. Schwabik, who read the manuscript and suggested several improvements. I am grateful to S. Schwabik who encouraged me and transformed the manuscript into the camera ready form. The research which resulted in publishing this book was supported by the grant No. 210/01/1199 of the Grant Agency of the Czech Republic.

Prague, March 2002

Jaroslav Kurzweil

CONTENTS

Preface

v

0. Introduction

1

1. Basic concepts and properties of ^-integration

9

2. Convergence

21

3. Convergence and locally convex spaces

32

4. An auxiliary locally convex space

42

5. £-integration

52

6. .M-integration

69

7. Noncompleteness

76

8. 5-integration

86

9. ^-integration

104

10. An extension of the concept of ^-integration

109

11. Differentiation and integration

116

References

135

List of symbols

137

Index

139

vn

0

INTRODUCTION

The approach to integration which is based on approximation of the integral by Riemannian sums is rather flexible. If the set of partitions which are used in the formation of Riemannian sums is rich then Lebesgue integration is obtained. On the other end of the spectrum a poor set of partitions leads to an integration which is called Henstock-Kurzweil and which is equivalent to Denjoy integration in the restricted sense and to Perron integration. In this book integrations are studied for various sets of partitions. If y is a set of partitions we denote by Py the set of primitives of 3^-integrable functions. For every 3? some sequences Fi G Py are called Ey- convergent IF"

to a limit FQ G Py, Fi —> FQ. Therefore there exists a unique locally convex topology ULc(^y) on Py which is the finest one among locally convex topologies T on Py with the property IF

that Fi —> Fo implies that Fi —> FQ in (Py,T). The problem whether (Py,Uic(^y)) is complete is crucial for this book; the answer depends on y. Let I = [a,b] C R. A set A = {(ti,Ai);i = 1 , 2 , . . . , A;} is called a partition in I if k G N and if (0.1) U G / f o r i = 1,2,...,*;, (0.2) Ai C I is a figure, i.e. a finite union of closed intervals, % =

i , 2 , . . . , K,

(0.3) figures Ai,Aj are nonoverlapping for i ^ j (i.e. \Ai D Aj\ = 0 for i ^ j where \E\ is the Lebesgue measure of 1

2

INTEGRATION BETWEEN L AND H - K

E C R). A is called a partition of / if (in addition) it

|J At = I.

(0.4)

Denote by C the set of partitions in / and by TiK, the subset of C which consists of A such that (0.5) Ai is a closed interval, i = 1, 2 , . . . , k, (0.6) U eAi,i

=

l,2,...,k.

+

Let ( : I -* R - A G £ is called (-fine if (0.7) At C (U - {(U),U + C(tx)) for i =

l,2,...,k.

f : I —» R is called Tf/C-integrable (£-integrable, respectively) if there exists j G R and for every £ G R + there exists ( : / -> R + such that

| 7 -^/(^)l^|| yk), k (=. N. The concept of ^-integration is an extension of the concept of 3^-integration and is studied in Chapter 10. Noncompleteness results are obtained if X = (<S(Ai), «S(A2),(Afc),... ), \ k £ A for k € N and if X = (K(u>i),K(uj2),K(ujk),...), uk € ft for k keN. If \k(a) = 2 a iora>0,X = (<S(A1),<S(A2),«S(Afc), • • •) then ^-integration is the integration which was introduced in [B

8

INTEGRATION BETWEEN L AND H - K

1996], if Lok(a) = 2~k for a > 0, X = ( f t ^ ) , ^ ^ ) , - ^ ^ , . . . ) then ^-integration is the integration which was studied in [B-Pf 1992]. Noncompleteness results in Chapters 9 and 10 are obtained by comparison with <S(A)-integration for a suitable A € A. 3^-differentiation is introduced in Chapter 11 and a general result concerning the relation of ^-differentiation and ^ - i n t e g ration is obtained. Chapter 11 is concluded by a specialization to £-integration. This Chapter will be closed by a list of four parts of the book which are not necessary for understanding the text which follows them: 1. Sections 4.7, 4.8, pp. 49-51 (a necessary and sufficient condition for completeness of (Py,ULc(Qy)) for a subclass of 3>'s); 2. Sections 5.8 - 5.11, pp. 62-68 (relations between convergences in (PC,ULC(QC)), in Qc and in Ec); 3. Chapter 6, pp. 69-75 (A^-integration); 4. Sections 8.9 - 8.12, pp. 91-103 (dependence of 5(A)integration on A).

1 BASIC CONCEPTS PROPERTIES OF

AND

^-INTEGRATION

1.1 N o t a t i o n . By R, R + , N, Z we denote the set of reals, the set of positive reals, the set of positive integers, the set of integers. For E C R, t G R let i n t E , c\E, \E\, d i a m E , d i s t ( t , £ ) be the interior of E, the closure of E, the outer Lebesgue measure of E, the diameter of E, the distance of t from E. Let N be the set of N C R such that |7V| = 0. By intervals we mean compact nondegenerate intervals in R, e.g. L = [c,d]. An open interval is denoted by (c, d) while [c, d), (c, d] are semiopen intervals. A figure is a finite union of intervals. T h e symbol #M stands for the number of elements of a finite set M. Let I — [a, b] be an interval. If K C I , then Iv(A') is the set of intervals L G K and Fig(A') is the set of figures A C K. We shall write Iv, Fig instead of Iv(J), Fig(J). Let K C I. A finite set A = {(i;,Aj); z = l , 2 , . . . , f c } , shortly A = {(i, A)} is called a partition in K if (1.1) R be ^-i-ntegrable and iet F : Fig —> R be its primitive. Then F is additive. Proof. Let KUK2 £ Fig, 1 ^ n K2\ = 0, j £ N. Let 6 £ D* correspond to / by Definition 1.3. There exist A^ = {(£, A)} £ y(Ki,Ki,6(j, •)), Ai being a partition of A^, z = 1,2. Then

(1.11)

|F(/v 2 )-5]/WI^I E be ^-integrable in I, F : Fig —> R being its primitive. Let 6 G -D* correspond to f by Definition 1.3 and let j G N, A = {(£, A)} G y(I,I,6(j,-)). Then (1.13)

|£(F(A)-/( ( A ) + ] T F ( c l L m ) . A

m€M

BASIC CONCEPTS

15

Therefore (cf. (1.16), (1.17)) | £ ( F ( A ) - f(t)\A\)\

< 2-> + # M • 2 - ' " - '

A

and (1.13) holds since p may be arbitrarily large. Let A1 = {(t,A)eA-F(A)-f(t)\A\>0}, A 2 = A \ Ai. By (1.13) we obtain £

\F(A) - / ( 0 | A | | = | £ 0 F ( A ) - f(t)\A\)\

Ai

< 2~\

Ai

J ] |F(A) - /(0|A|| = | £(F(A) ~ f(*)\A\)\ < 2"J" A2

A2

so that (1.14) holds. The proof is complete 1.8 Definition. Let H : Fig -* R, s € J. if is called J continuous at s if for every e £ R + there exists < e , a < % + 2 , 5 ) , 2 a | / ( s ) | < 2"-''- 1 . By (1.14) we have J2 \F(A)\

< •2-^-1 + \f{s)\J2

A

\A\ < 2-i-1

+ \f(s)\2 R. Py is a vector space, Py C AdC (cf. Theorem 1.6, Theorem 1.9). 1.11 D e f i n i t i o n . Let H : Fig -> R or H : Iv - • R, s <E / , (3 £ R. H is called differentiable at S and /? is the derivative of H at s if for every e <E R + there exists R be y-integrable be its primitive. Then (1.18) F(t) exists almost everywhere, erywhere, f is measurable.

F(t) = f(t)

and let F almost

ev-

(1.18) is a consequence of [K 2000], Theorem 1.17 since 7i)C C y and / is 7Y/C-integrable. 1.13 L e m m a . Let h : I - • R, N G M, e G R+. Then exists ( : I —•> R+ such that

there

£>(0PI<e A

forA = {(t,A)}€£(I,JV,C). # m i (cf. [K 2000], Lemma 1.15). For i G N there exist open sets d CR such that N C Gt, \G{\ < e2~2i. Put E0 = 0, Et = {t G iV; |fc(*)| < 21}. L e t ( : I ^ R + fulfil (*-((*),*+((*)) C Gt for t e Ei\ Ei-i, i G N. Then £ does the job. 1.14 T h e o r e m , (i) Let / , # : I —> K be ^-integrable, (1.19)

(3>) / / ( s ) d 5 = (y) Ja

I g{s)ds for < G / . •/ a

Then f = g almost everywhere. (ii) Let / , g : / —> R. Assume £ba£ / is [V-mtegrabie and / = g almost everywhere. Then g is y-integrable and (1.19) holds. Proof, (i) is a consequence of Theorem 1.12 and (ii) follows by Lemma 1.13. 1.15 L e m m a . Let K G Fig, / : K -»• R, F : Fig(A') -> R. Assume that F is additive. Then the following conditions are equivalent: (1.20) / is y-integrable

and F is its

primitive,

18

INTEGRATION B E T W E E N L AND H - K

(1.21) there exists 9 £ D* such that

Y,\F(A)-f(t)\A\\ A

for j £ N, A = {(t, A)} £ y(K, K, 0(j, •)), (1.22) there exist M £ M and r\ £ D* such that (i)

£|F(A)-/(*)|A|| R be additive and fulfil (2.14) and (2.15). Then F is the y-primitive of f.

Proof. Let M be defined by (2.15). By Lemma 2.7 there exists £ € D* such that

2^/jr,^ *(M)

z

e

for j € N, 0 = {(t, A)} € y(I,M,Z(J,-))Hence (1.22) holds if r](j,t) < min{0(j + l,t),£(j + 1,*)} and Lemma 2.8 is a consequence of Lemma 1.15. T h e proof is complete. 2.9 T h e o r e m .

U QyW) = py9 R7 Go '• Fig —> EL Assume that (2.17)

G 8 (A) -»• G0(A)

for A e Fig,z -> oo.

T i e n there exists go : i" —• R. such that (2.18)

^i —• ^o ha measure for i —> oo.

Note. The theorem has several consequences: (2.19) there exists a subsequence i(k), k £ N and Mo (E TV such that 9i(k)(t) -* #o(0 for k -+ oo a.e., (2.20) G 0 is additive (by (2.17), since G; are additive), (2.21) Go e Qy(0) (since we may write G ,-(£), oo by (2.17), (2.19) after having made a suitable choice of M ) , (2.22) Qy(8) is closed in AdC (cf. (2.21), (2.17), (2.19)), (2.23) Gi(fc) —>• Go (since G m e Qy{6) for m = 0 , 1 , 2 , . . . and (2.19) holds, there exist M G Af, rj G D* such that (2.9) (i) and (iii) hold. Moreover, (2.9) (ii) follows from (2.14) by Lemma 1.13 so that (2.23) holds by Lemma 2.3). Proof. In order to prove (2.18) it is sufficient to show that the sequence gi is Cauchy in measure. Let us suppose the opposite, i.e. that there exists a 6 R + such that for every r € N there are p = p(r), q = q(r) such that p,q > r and (2.24)

\{tel;

\gp(t)-gg(t)\>a}\>2a.

CONVERGENCE

27

Let h £ N be chosen such that 2~h+2b-^}.

Then J 5 ^ C Xi forfc< / and (Jj.^Xfc = / so that lim \Xk\ = b — a. k—>oo

Find k eN such that \Xk\ > b — a — a. Denote EVA = {te I ; |flf„(*)-*}By Theorem 1.12 2 00. Denote by Qy the set of couples ((F 8 , i G N ) , F 0 ) fulfilling (2.31) and (2.32). Qy is called a convergence and the sequence Pi, z G N is said to be convergent

to FQ in Qy, Ft —> Fo.

2.15 L e m m a . Let F{ ^U F0, Gt ^ Then (2.33)

(2.34)

(2.35)

G0, a G M.

Pi(jfe) — • Po for any subsequence

i^ + G ^ F o

+ Go,

a f ^ a F o .

i(k), k = 1, 2, 3 . . . ,

CONVERGENCE

31

Proof. By Definition 2.14 there exist 771,772 £ D* such that FleQy(r]l),GleQy(r]2)

for i = 0 , 1 , 2 , . . .

and 11-ft - -Po| I sup —> 0, ||Gi - Go||sup —>• 0 for z —• 00. Obviously, (2.33) is valid. (2.34) holds since there exists 773 £ D* (cf. Lemma 2.13) such that Ft + Gi GQyM

for i = 0 , 1 , 2 , . . .

and H^ + G i - F o - G o l l s u p - O . Let a £ R, k £ N, | a | < 2*. Making use of Lemma 2.13 repeatedly we conclude that there exists 0 £ D* such that 2*ft GQy(O)

for i = 0 , 1 , 2 , . . . .

Hence a f t £ # y ( 0 ) for z = 0 , 1 , 2 , . . . and (2.35) holds since \\aFi — a-Fo||sup —• 0. T h e proof is complete. 2.16 T h e o r e m , (i) If F{ ^

F0, then ft ^

(ii) If Fi — • Fo then there exists a subsequence such that Fi(k) ^

F0. i(k), k £ N

FQ.

Proof. Let Fi —^» -Fo- By Definition 2.2 there exist / m , m = 0 , 1 , 2 , . . . and 6 £ D* such that (2.6) and (2.7) hold. Moreover, there exists 9 £ D* such that

0(j,t) fc. Let T\,T2 be topologies on F . 7i is called ./mer than T2 if 7i D T 2 . From now on let Y be a vector space over R. Let T be a topology on F . If y G F , t/ G T implies that y + U C T, then T is called invariant with respect to shifts. 32

CONVERGENCE AND LCS

33

Let 53 C 2 y fulfil (3.4)

(3.5)

(3.6)

03 ^ 0, 0 i 03,

every V from 03 is radial, circled and convex,

for Vi, V2 G 03 there exists V3 G 03

such that v3 c Vi n v2, (3.7)

(3.8)

for V G 03 there exists Vi G 03 such that 2Vi C V,

for x G F , x 7^ 0 there exists V G 03 such that x ^ V.

Let V(03) be the set of U C Y such that x G U if and only if x + V C U for some V G 03. V(03) is a topology on F which is invariant with respect to shifts and Hausdorff. V(03) is called the topology induced by 03. (Y,T) is called a locally convex space if there exists 03 such that T = V(03). Let r : Y -+ [0, oo). If (3.9)

(3.10)

r(x + y) < r(x) + r(y)

for

i,t/6F,

r ( ^ x ) = |)u|r(x) for x G F, /u G R

then r is called a seminorm. Let i J b e a set of seminomas. Assume that for every x G F there exists r G -R such that r(x) > 0. The set (3.11)

{x G F ; r,(x) < £ for i = 1 , 2 , . . . , k}

INTEGRATION BETWEEN L AND H - K

34

where k € N, r; G R, e G R + is convex, circled and radial. Let 2G be the set of sets (3.11). 2U fulfils (3.4) - (3.8) and V(2U) is called the topology induced by R. Let P* C AdC be a vector space. Let A be a set the elements of which are couples ((Fi,i G N),Po) where Fm G P* for m = 0 12 V,

-L, - : , . . . •

We shall write (3.12) Ft - ^ F0 instead of ((Fi,i G N ) , P 0 ) G A and assume (3.13) i f P , - ^ P 0 , G 4 - ^ G 0 , a , / ? G E t h e n ( a ^ + / ? G O ^ (aF0+/3G0),

aFi -^

aF0,

(3.14) if Ft - ^ P 0 then ||P, - P 0 | | s u p - • 0. A is called a convergence on P* and the sequence Pj-,i G N is said to be convergent

to F0 in A, Fi

A

> P0.

3.2 L e m m a . Let P* C P2* C AdC, Pi*,.P2* being spaces. Let A* be a convergence on P ? , i = 1,2. T i e n (3.15)

Ax C A 2 iff P, - ^ P 0 implies Ft-^

vector

F0.

This is just a reformulation of (3.12). 3 . 3 D e f i n i t i o n . Let P * C AdC be a vector space, let A be a convergence on P * and let T be a topology on P * . T is called tolerant to A if Ft - ^ F0 implies Ft -> P 0 in

{P*,T).

3.4 N o t a t i o n . Denote by 7^ up the topology on AdC which is induced by the norm || • ||Sup- Let P * C AdC be a vector space and let A be a convergence on P * . By U L C ( A ) let us denote the set of topologies T on P * such that (3.16) (3.17)

(P*,T)

is a locally convex space, y

is tolerant to A.

Note. U L C ( A ) / 0 since { 0 , P * } G U L C ( A ) .

CONVERGENCE AND LCS

35

3.5 L e m m a . Let P* C AdC be a vector space and let A be a convergence on P*. Then there exists ULC(A) G U L C ( A ) such that (3.18)

W L C (A) is finer than any T G U L c ( A ) .

3.6 N o t e . Uic{A) is unique. Moreover, ULC(A) dorff since it is finer than T SU p|p*.

is Haus-

Proof. If T G U L C ( A ) then (P*,T) is a locally convex space and there exists 03 G 2 P * fulfilling (3.4) - (3.8) such that T = V(03). For a finite set 7i = V(03i), T2 = V(03 2 ), . . . , Tk = V(03fc) G U L C ( A ) and Vi G 3 such that F G ( 1 - £ ) F SO that F + — F C ( 1 - — )V. m m Let / € N, 2' > m. Making use of (3.7) repeatedly we conclude that there exists V £ 53 such that 2lV C V. Hence

i>c2 _ 'v c — v m and F + VC(1-—)V

m

CV0

so that Vo is open. The proof is complete. 3.9 T h e o r e m . Let T be a topology on Py. tolerant to Qy if and only if it is tolerant to Ey.

Then T is

CONVERGENCE AND LCS

Corollary. ULC{Ey)

= VLC(Qy),

ULC(Ey)

37

=

ULC(Qy).

Proof. If T is tolerant to Qy then it is tolerant to Ey by Theorem 2.16 (i). Assume that T is tolerant to Ey and that there exist Fm G Py, m = 0 , 1 , 2 , . . . such that Fi —> FQ but the sequence Ft, i G N is not convergent to F0 in ( P y , T ) . Hence there exist U G T and a subsequence i(k), k G N such that FQ £ U and (3.20) However, F^)

F i ( f e ) £ U for fc G N. ~~^ -Fo by (2.33) and Theorem 2.16 (ii) implies

that there exists a subsequence &(/), / G N such that Fj(fc(;)) —> FQ. Therefore there exists IQ G N such that F;(j.(/)) £ U for I > IQ, which contradicts (3.20). This contradiction proves that T is tolerant to Q y . T h e proof is complete. 3 . 1 0 L e m m a . For a G R + p u t 5 ( a ) = {F G AdC; | | F | | s u p < a}. Let 0 G 17 G ^ L c ( Q y ) , 0 £ D*. Then there exists a G R + such that Qy(0) D £( < 2~\ Hence Fi > 0 which is impossible since ULci'Q'y ) is tolerant to Q y . This contradiction makes t h e proof complete. 3.11 L e m m a . For E C AdC denote by c o n v F the convex hull of E. LetOeUe ULC(Qy)Then there exists ( : D* -> + R such that conv | J Q y ( 6 ) n B(C(6)) C U. seD*

INTEGRATION BETWEEN L AND H - K

38

Proof. Since 0 G U € WLC(Q;>;) and WLC(Q3>) = V(2J) for some R + such that | J Qy(6) n B(((6))

C Vo

and conv \J Qy{8) n # ( ( ( £ ) ) C Vo C V C f/. seD* The proof is complete. 3 . 1 2 L e m m a . For ( e D* define V>(C) : N x J -+ R+ by

(3.21)

^(C)0',0 = |C0" + 1,0-

Then (3.22)

il>:D*^

(3.23) V>(C) = * iff <M,t) = 26(j-l,t) and ((l,t) (3.24)

2Qy(C)

D*,

for j =2,3,4,...,t

> ((2,t)

C Qy(H0)

el

for t e I, for

C € #*•

Proof. (3.22) and (3.23) are immediate consequences of (3.21). Let F e Qy(0- Then

CONVERGENCE AND LCS

39

for j G N, A = {(t, A)} G y{I, I, CO', •)) and there exist M G AT and / : I —> R such that ^|F(A)-/(t)|A||" 1 + £

E. 4.2 T h e o r e m . Assume that (4.3)

P y C Ty,

(4.4)

Ty\py

(4.5)

TKF

(4.6)

for A' e Iv

is tolerant to

Qy,

£ Py for K € Iv ,F £ P y , there exists

I I F / c P b , , < K\\F\\y,„

K£ R

such that

FePy.se

Then (4.7)

ULC{Qy)

= Ty\Py.

T h e proof will be performed in several steps.

I.

INTEGRATION BETWEEN L AND H - K

44

4 . 3 L e m m a . Let K(i) = [ci,dt] G Iv, K(i + 1) C K(i) zeN, s e l , f]ienK(i) = {s}. Let Ft G Py fulfil

for

(4.8) Fi(A) = 0 if A G Fig , A C [a,ct] U [d,-, 6], « G N, (4.9) for e G R+ there exists a G R + such that

E l*K^)l ^ £ A

fori€N,A = {(s,A)}6y(I,{s}, R and Mt e JV for z G N such that

(4.10)

JX^I^+^J^ A

A

u

'

y

for i , ; G N, A = {(*,4)} G ? ( / , J , * ( j , •)),

(4.11)

^lF^)-/«(t)lAH^2"i A

for i,JGN, A = { ( t , A ) } G ^ J \ M , , ^ 0 V ) ) . Without loss of generality we may assume that (4.12) fi(t) = 0 for a < t < a, di < t < b and t = s, i G N, (4.13) Sk(j,t) < 6i(j,t) for i,j,ke N,k>i,tel.

AN AUXILIARY LCS

Put (4.14) ui(t) = dist(«, K(i)) for i e N, t e I and define 9 as follows: (4.15) 0U,t)=u1{t)

for j G N , t G / \ / i ( l ) ,

9(j,t) =min{u)t+1(t),6t(j foriJtN,

+ i + l,t)}

f € A ^ ) \ A ' ( « + l),

9(j,s) is so small that ^|Fi(A)|