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Problem Books in Mathematics

Series Editor: P.R. Halmos

Unsolved Problems in Intuitive Mathematics, Volume I:Unsolved Problems in Number Theoryby Richard K. Guy1981. xviii, 161 pages. 17 illus.

Theorems and Problems in Functional Analysisby A.A. Kirillov and A.D. Gvishiani1982. ix, 347 pages. 6 illus.

Problems in Analysisby Bernard Gelbaum1982. vii, 228 pages. 9 illus.

A Problem Seminarby Donald J. Newman1982. viii, 113 pages.

Problem-Solving Through Problemsby Loren C. Larson1983. xi, 344 pages. 104 illus.

Demography Through Problemsby N. Keyfitz and J.A. Beekman1984. viii, 141 pages. 22 illus.

Problem Book for First Year Calculusby George W. B/uman1984. xvi. 384 pages. 384 illus.

Exercises in Integrationby Claude George1984. x. 550 pages. 6 illus.

Exercises in Number Theoryby D.P. Parent1984. x. 541 pages.

Problems in Geometryby Marcel Berger et al.1984. viii. 266 pages. 244 illus.

Claude George

Exercises inIntegration

With 6 Illustrations

Springer-VerlagNew York Berlin Heidelberg Tokyo

Claude George Translator

University de Nancy I J.M. ColeUER Sciences Mathematiques 17 St. Mary's MountBoite Postale 239 Leybum, North Yorkshire DL8 5JB54506 Vandoeuvre les Nancy Cedex U.K.France

EditorPaul R. HalmosDepartment of MathematicsIndiana UniversityBloomington, IN 47405U.S.A.

AMS Classifications: OOA07, 26-01, 28-01

Library of Congress Cataloging in Publication DataGeorge, Claude.

Exercises in integration.(Problem books in mathematics)Translation of: Exercices et problemes d`int6gration.Bibliography: p.Includes indexes.

1. Integrals, Generalized-Problems, exercises,etc. I. Title. II. Series.QA312.G39513 1984 515.4 84-14036

Title of the original French edition: Exercices et problemes d'integration,© BORDAS, Paris, 1980.

© 1984 by Springer-Verlag New York Inc.All rights reserved. No part of this book may be translated or reproduced in any formwithout written permission from Springer-Verlag, 175 Fifth Avenue, New York,New York, 10010, U.S.A.

Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia.Printed in the United States of America.

987654321

ISBN 0-387-96060-0 Springer-Verlag New York Berlin Heidelberg TokyoISBN 3-540-96060-0 Springer-Verlag Berlin Heidelberg New York Tokyo

Introduction

Having taught the theory of integration for several years at

the University of Nancy I, then at the Ecole des Mines of the same

city, I had followed the custom of the times of writing up de-

tailed solutions of exercises and problems, which I used to dis-

tribute to the students every week. Some colleagues who had had

occasion to use these solutions have persuaded me that this work

would be interesting to many students, teachers and researchers.

The majority of these exercises are at the master's level; to them

I have added a number directed to those who would wish to tackle

greater difficulties or complete their knowledge on various points

of the theory (third year students, diploma of education students,

researchers, etc.).

This book, I hope, will render to students the services that

this kind of book brings them in general, with the reservation

that can always be made in this case: that certain of them will

be tempted to look at the solution to the exercises which are put

to.them without any personal effort. There is hardly any need to

emphasize that such a use of this book would be no benefit. On

the other hand, the student who after having worked seriously

upon a problem, seeks some pointers from the solution, or compares

it with his own, will be using this work in the optimal way.

V

vi INTRODUCTION

Teachers will find this book to be an important, if not ex-

haustive, list of exercises, certain of which are more or less

standard, and others which may seem new.

I have also noted (and this is what led me to edit these sheets)

that from one year to another one sometimes forgets the solution

of an exercise and that one has to lose precious time in redis-

covering it. This is particularly true for those solutions of

which one remembers the heuristic form but of which the writ-

ing up is delicate if one wishes to be clear and precise at the

same time. Now, if one requires, quite rightly, that students

write their homework up correctly, then it is befitting to sub-

mit impeccable corrections to them, where the notations are jud-

iciously chosen, phrases of the kind "it is clear that ... " used

wittingly, and where the telegraphic style gives way to concise-

ness. It is often the incorporation of these corrections which

demands the most work; I have therefore striven to take pains

with the preparation of the proposed solutions, always remaining

persuaded that perfection in this domain is never attained. If

this book encourages those who have to present (either orally or

in writing) correct versions of problems to improve the version

they submit, the object I have set myself will be partly realised.

In this book researchers will find some results that are not

always treated in courses on integration; they are either proper-

ties whose use is not as universal as those which usually appear

and which are therefore found scattered about in appendices in

various works, or are results that correspond to some technical

lemmas which I have picked up in recent articles on a variety of

subjects: group theory, differential games, control theory, prob-

ability, etc., ... .

In presenting such a work it is just as well to make explicit

those points of the theory that are assumed to be known. This is

the object of the brief outline which precedes the eleven chapters

of exercises.

INTRODUCTION vii

In view of the origin of this book, it is evident that I took

as a reference point the course that I gave at the time. After

having taught abstract measure theory one year, I opted the next

for a course expounding only the Lebesgue integral. This is not

the place to discuss the advantages and inconveniences of each of

the two points of view for the first year of a master's programme.

I will say only that I have always considered the course that I

gave to be more a course in analysis in which it is decided to

use the Lebesgue integral than as a dogmatic exposition of a par-

ticular theory of integration. The choice of exercise reflects

this attitude, especially in the emphasis given to trigonometric

series, thereby paying the hommage due to the theory which is the

starting point of the works of Cantor, Jordan, Peano, Borel, and

Lebesgue. From this it results that, except for the seven exer-

cises of Chapter 2 concerning a-algebras, all the others deal with

Lebesgue measure on]Rr. The advantage that has to be conceded to

this point of view is that it avoids the vocabulary of abstract

measure theory, which constitutes an artificial obstacle for those

readers who might not yet be well versed in this theory. As for

students who might have followed a more sophisticated course, I

can assure them that by substituting du for dx and u(E) fore

meas(E) they will essentially rediscover the problems as they are

commonly put to them, except for pathological examples about mea-

sures that are not a-finite and the applications of the Radon-

Nikodym Theorem. Furthermore, on this latter point the more per-

spicacious amongst them will not fail to see that the chapter

treating the relationships between differentiation and integra-

tion is not foreign to this theorem. Truthfully, there is another

point that is not tackled in this book, namely the matter of Four-

ier transforms of finite positive measures and Stone's Theorem,

which to my mind is better suited to a course on probability.

As was mentioned above, numerous exercises are devoted to trig-

onometric series, which provides an important set of applications

Viii INTRODUCTION

of Lebesgue's theory. This has led me to include some exercises

on series, summation processes, and trigonometric polynomials.

Other exercises use the theory of holomorphic functions. In

particular, some results of the PhrUgmen-Lindeltff type arise on

two occasions; in each instance I have given its proof under the

hypotheses that appear in the exercise. Quite generally, I have

included in the solutions, or in an appendix to them, the proofs

of certain points of analyis, topology, or algebra which students

may not know.

I have chosen to make each solution follow immediately after

the corresponding problem. The other method, which consists of

regrouping the former in a second part of the work, seemed to me

(from memories I have retained from my student days) much less

manageable, especially when the problem is long, for it then be-

comes necessary to return often to the back of the book in order

to follow the solution.

I find it difficult to cite the origin of these exercises.

Many are part of a common pool of knowledge, handed down, one

might say, in the public domain. Others are drawn from different

classic works where they are proposed without proof or followed

by more or less summary indications (in this respect it is inter-

esting to note that in forcing oneself to write down the solutions

one discovers a certain number of errors -just as many in the

questions as in the suggestions offered). Certain of the exer-

cises in this book were communicated to me orally by colleagues;

I would thank them for their help here. Lastly, others are, as

I have already said, lemmas found here and there, and which I

have sometimes adapted.

Table of Contents

INTRODUCTION ... ... ... ... ... ... ... ... V

CHAPTER 0: OUTLINE OF THE COURSE ... ... ... ... ... 1

CHAPTER 1: MEASURABLE SETS ... ... ... ... ... ... 37

(Exercises 1.1 1.21)

CHAPTER 2: a-ALGEBRAS AND POSITIVE MEASURES ... ... ... 79

Exercises 2.22 - 2.28)

CHAPTER 3: THE FUNDAMENTAL THEOREMS. ... ... ... ... 89

(Exercises 3.29 - 3.72)

CHAPTER 4: ASYMPTOTIC EVALUATION OF INTEGRALS... ... ... 177(Exercises 4.73 - 4.78)

CHAPTER 5: FUBINI'S THEOREM... ... ... ... ... ... 199

(Exercises 5.79 - 5.99)

CHAPTER 6: THE LP SPACES ... ... ... ... ... ... 225

(Exercises 6.100 - 6.125)

CHAPTER 7: THE SPACE L2. ... ... ... ... ... ... 285

(Exercises 7.126 - 7.137)

CHAPTER 8: CONVOLUTION PRODUCTS AND FOURIER TRANSFORMS ... 325

(Exercises 8.138 - 8.162)

CHAPTER 9: FUNCTIONS WITH BOUNDED VARIATION: ABSOLUTELYCONTINUOUS FUNCTIONS: DIFFERENTIATION ANDINTEGRATION (Exercises 9.163 - 9.173)... ... 405

ix

x TABLE OF CONTENTS

CHAPTER 10: SUMMATION PROCESSES: TRIGONOMETRIC POLYNOMIALS.. 429(Exercises 10.174 - 10.184)

CHAPTER 11: TRIGONOMETRIC SERIES ... ... ... ... ... 451(Exercises 11.185 - 11.230)

ERRATUM TO EXERCISE 3.45 ... ... ... ... ... 545

BIBLIOGRAPHY ... ... ... ... ... ... ... ... 547

NAME INDEX.. ... ... ... ... ... ... ... ... 549

CHAPTER 0

Outline of the Course

0.1 a-ALGEBRAS AND MEASURES

DEFINITION: A family A of subsets of a set X is called a a-ALGEBRA

("sigma algebra") if 0 e A, and if A is closed under complementation

and countable union.

From this it follows that the set X itself belongs to the a-alge-

bra A, and that the a-algebra A is closed under countable intersec-

tion. For two sets A,B e A let us denote A - B = {x:x a A,x $ B};

then we have (A - B)e A.

The smallest a-algebra containing the open sets of ]R is the

a-algebra of BOREL SETS of ]R ; this a-algebra is also the small-

est a-algebra which contains the closed (resp. open) rectangles

of]R .

DEFINITION: A (positive) MEASURE on a a -algebra A is a mapping u

of A into [0,oo] such that if E is the disjoint union of a sequence

of sets En e A, then u(E) = I u(En).

It follows that u(o) = 0, and then, if E is the union (not

necessarily disjoint) of the sets En, U(E) 5 L u(En). An equi-

valent definition is the following: If E is the union of a fin-

1

2 CHAPTER 0: OUTLINE

ite number of Ei's, each of which is in A and which are pairwise

disjoint, then p(E) = p(E1) + + p(Ep); and furthermore p(A) _

limu(An) when A is the union of an increasing sequence of sets An

of A. If p is a measure and A is the intersection of a decreas-

ing sequence of sets An e A and if u(A1) < -, then p(A) = limp(An)

There exists one and only one positive measure v on the a-alge-

bra of Borel sets of ]R such that, for every rectangle P, its

measure v(P) is equal to the volume of P.

DEFINITION: A set E of]R is called a NEGLIGEABLE SET if there

exists a Borel set A such that E C A and v(A) = 0.

This definition is equivalent to the existence, for every e> 0,

of a sequence of rectangles covering E, the sum of the volumes of

which is less than c. A countable union of negligeable sets is

negligeable, and every affine sub-manifold of iRp that is of dimen-

sion less that p is negligeable.

DEFINITION: A set of ]R is called a LEBESGUE MEASURABLE SET (or

simply a MEASURABLE SET) if it belong to the smallest a-algebra

containing the Borel sets and negligeable sets of]R1.

In order that E C ]R be measurable it is necessary and suffic-

ient that there exist the Borel sets A and B such that A C E C B

and v(B - A) = 0; upon then setting meas(E).= v(A) one unambig-

uously defines a positive measure on the a-algebra of Lebesgue

measurable sets of ]Rp. This measure is called the LEBESGUE MEA-

SURE ON I(. A set is negligeable if it is measurable and of

(Lebesgue) measure zero. This is why one also uses the expres-

sion SET OF MEASURE ZERO to denote a negligeable set.

The Lebesgue measure is invariant under translation as well

as under unimodular linear transformations (i.e., those with de-

terminant equal to ±1). A homothety of ratio A multiplies the

Lebesgue measure by JAJp (where p is the dimension of the space).

OF THE COURSE 3

DEFINITION: If A is a o-algebra of subsets of X and B is a Q alge-

bra of subsets of Y, a mapping f:X + Y is said to be an A -

B-MEAS-URABLE MAPPING if f 1(B) e A for a Z Z B e B.When Y= Ilzp and B isthe a-algebra of Borel sets, one says, simply, that f is an A-MEAS-

URABLE MAPPING. In this case the definition is equivalent to re-

quiring f_1(V) eA for every open set V of Y. Furthermore, when

X = n2q, the mapping f is said to be a BOREL MAPPING or a LEBESGUE-

MEASURABLE MAPPING according as A is the a-algebra of Borel sets

or the Lebesgue-measurable sets of X.

If f:X -;Ill, in order that f be A-measurable it is sufficient

that (f < a) = {x:f(x) < a} e A for all a em (and even for a e

This condition is taken as the definition of the A-measurability

of an ARITHMETIC FUNCTION, that is to say of a mapping of X into

[-co,+m] =3-R. If f is an A-measurable mapping of X into Iltp and g

a Borel mapping of Ilzp into zzq, then gof is A-measurable. Let us

note that every continuous mapping of Ilzp into Ilzq is Borel. If

(fn) is a sequence of A-measurable arithmetic functions, the func-

tions supfn,inffn,limsupfn,liminffn are also A-measurable.

DEFINITION: A function is called a SIMPLE FUNCTION (with respect

to the a-algebra A) if it is a linear combination of characteristic

functions of sets of the a-algebra A.

For every A-measurable positive arithmetic function f there ex-

ists an increasing sequence of positive simple functions which

converges to f at every point of X.

DEFINITION: A property holding on the points of a set A of7Rp is

said to be true ALMOST EVERYWHERE ON A SET A if the set of points

of A for which this property is not satisfied has measure

zero.

If f and g are two mappings from z into zzq (or m) such that

f is measurable and f = g almost everywhere, then g is measurable.


This allows the notion of measurability to be extended to func-

tions that are defined only almost everywhere.

DEFINITION: A function defined on]RP is called a STEP FUNCTION if

it is a linear combination of characteristic functions of rect-

angles of]RP.

Every measurable arithmetic function on]R is the limit almost

everywhere of a sequence of step functions.

THEOREM: (Regularity of the Lebesgue Measure): For every measur-

able set E ofiRP one has:

sup{meas(K):K compact K C E};

meas(E) =

inf{meas(V):V open V D E}.

THEOREM: (Egoroff): Let X be a measurable set of]R such that

meas(X) < co and (fn) a sequence of measurable functions such that

fn- f almost everywhere on X. For every e > 0 there exists a

measurable set A C X such that:

(i): meas(X - A) < ci

(ii): fn -> f uniformly on A.

0.2 INTEGRATION OF MEASURABLE POSITIVE FUNCTIONS

NOTATION: If cp is a simple function on Min that takes the positive

values a1,...,ap on the (disjoint) measurable sets Al....)Ap, we

set

I q,(x)dx = 9 _ aimeas(Ai),n i=1

OF THE COURSE 5

with the convention that a.(+-) or 0 according as a > 0 or

a = 0.

DEFINITION. With the above notation, if f is a positive meas-

urable arithmetic function on ]Rn there exists an increasing

sequence ((pi) of positive simple functions which tends towards f

at every point. One then sets:

J f(x)dx = if = limf . .

This element of [0,+-]=]K +, which does not depend upon the sequence

(Ti) selected, is called the (LEBESGUE) INTEGRAL of f on7Rn.

This (Lebesgue) integral possesses the following properties

(where f and g denote measurable positive arithmetic functions):

PROPERTY (1): If f = g almost everywhere, then if = Jg;

PROPERTY (2): Jf = 0 if and only if f = 0 almost everywhere;

PROPERTY (3): if < - implies f < m almost everywhere;

PROPERTY (4): f 4 g almost everywhere implies if 1< Jg;

PROPERTY (5): J(f + g) = if + fg;

PROPERTY (6) If A e]-R+., then JAf = AJf.

One can prove the following two fundamental results:

THEOREM: (Lebesgue's Monotonic Convergence): If (fn) is an in-

creasing sequence of measurable positive arithmetic functions,

then

Jlimf = limif-'n n n


LEMMA: (Fatou): If (fn) is a sequence of positive arithmetic func-

tions, then

Jliminffn " liminfn n

Ifn.

These two essential properties of the Lebesgue integral are

equivalent to the following statements: Let (fn) be a sequence

of positive arithmetic functions:

(a): f(I fn) = E (Jfn);

(b): If fn -* f almost everywhere, and if there exists A eIt+

such that

Jfn 5 A for all n,

then

Jf , A.

Property (1) allows the definition of the integral to be ex-

tended to measurable arithmetic functions that are defined only

almost everywhere.

And lastly:

NOTATION: If f is a measurable positive arithmetic function, and

if E is a measurable set of ]R , we set

fEf(x)dx=JE =fiv,

where 1lE denotes the CHARACTERISTIC FUNCTION of E.

The mapping E + J f defines a positive measure on the 'a-algebraE

OF THE COURSE 7

of (Lebesgue) measurable sets.

0.3 INTEGRATION OF COMPLEX MEASURABLE FUNCTIONS

NOTATION: For every real function u, we set:

u+ = sup(u,O) _ I(luI + u),

u- = sup(-u,0) _ i(lul - u),

so that

u = U+ - u_,

lu l = u+ + u-,

u+u- = 0.

Let f be a complex measurable function on7Rn, and let u and v

be its real and imaginary parts. The function f is measurable

if and only if u+U_,v+,v- are measurable.

DEFINITION: f is (LEBESGUE) INTEGRABLE ON]R if it is measurable

and if

11fl <

The INTEGRAL is then defined by setting

Jf = Ju+ - Ju + iJv - iJv,

(which has a meaning, because u+,u_,v+,v_ are majorised by lfl).

The Lebesgue integral possesses Properties (1) and (5) of the

preceding Section; it also possesses Property (4) when f and g

are real, as well as Property (6) with A e T. The sum, and the

pointwise maximum and minimum of a finite number of integrable


functions are integrable. If f is integrable, then

ff1 <J1f11

with equality holding only if there exists a e 0 such that f =

alfl almost everywhere.

The two essential properties of the Lebesgue integral are the

following:

THEOREM: (Lebesgue's Dominated Convergence): Let (fn) be a sequence

of integrable functions that converges to f almost everywhere. If

there exists a measurable positve arithmetic function g such that

lfnl 5 g for all n, and Jg < W,

then f is integrable, and

Jf= limn Jfn.

THEOREM: (Term by Term Integration of Series of Functions): If

(un) is a sequence of integrable functions such that

I JUj

<

n

then the series

u(x) = I un(x)n

is absolutely convergent for almost all x, the function u (which

is defined almost everywhere) is integrable, and

Ju = fUnn

OF THE COURSE 9

If f is integrable and if E is a measurable set, we again set

JE = JLEf

(the integral of f over E). In fact this integral depends only

upon the values of f on E and can be defined whenever f itself is

only defined on the set E; it suffices, for example, that the

function obtained by extending f by zero outside E be integrable

over]R". In this case one says that f is INTEGRABLE OVER E. All

the properties of the Lebesgue integral over3Ru extend to this

case. Furthermore, if f is integrable over E then it is integrable

over every measurable set contained in E, and if E is the disjoint

union of a sequence (En) of measurable sets,

fEf

= L JE f.nn

Similarly, if E is the union of an increasing sequence (En) of

measurable sets,

f.J f = l imfEE n

n

This formula is still valid if E is the intersection of a de-

creasing sequence (En) of measurable sets and if f is integrable

over E1.

In the case of integration on ]R, it is convention to write

when I is an interval with endpoints a and b(-o 4 a 4 b 4+-),

and f is either a measurable positive arithmetic function on I,

or a complex integrable function on this interval. When a > b


we write

when f is integrable over [a,b], and a "Chasles' Formula" can then

be written. If -- < a < b < +-, then f is Lebesgue-integrable

over [a,b] whenever it is Riemann-integrable over this interval,

and the two integrals are equal. However, when the interval is

infinite the Lebesgue integral only constitutes an extension of

the notion of absolutely convergent (generalized) Riemann integral.

The GENERALIZED (or SEMI-CONVERGENT) LEBESGUE INTEGRALS can be

defined in the following way. For example, let us assume that f

is (Lebesgue) integrable over every interval [0,M], 0 : M <

one then sets

Mf

J_f

M}oj0

when this limit exists. In this respect let us note the follow-

ing Proposition:

PROPOSITION: (Second Mean Value Formula): If f is a decreasing

positive function on [a,b], and g an integrable function on this

interval, then

IJbfgl : f(a) sup IJxgl.

a a,<xsb a

To close this Section let us indicate that if f is a mapping

from3RP into3Rq of which the q coordinates are integrable, the

integral of f is the element 0f] of which each component is

equal to the integral of the corresponding component of f. Every-

thing that has been said above remains valid when the absolute

OF THE COURSE 11

value is replaced by a norm on U .

0.4 FUBINI'S THEOREM

THEOREM: (Fubini) : Let X = Iltp, y = ]R then the formula

jjXxf (x,y)dxdy = fXdxjf(x,y)dy =fy

dyfXf(x,y)dx

is valid in each of the following two cases:

(1): f is a measurable positive arithmetic function on X x Y;

(2): f is an integrable function over X x Y.

Amongst other things the validity of the formula means (accord-

ing to the case) that for almost all x e X the function y Fa f(x,y)

is measurable (resp., integrable) on Y, and that the function

x ' J f(x,y)dy, defined almost everywhere on X, is measurable

X(resp., integrable) on X.

In particular, in order to have the rule of "interchangeability

of the order of integration" it suffices to be assured, when f is

a measurable complex function on Xx Y, that

(x,y)ldy <JJ1If

Whenever f is positive and measurable this interchange is al-

ways legitimate.

Certain proofs of Fubini's Theorem use the following Lemma,

which is interesting in its own right:

LEMMA: In order that a set have measure zero it is necessary and

sufficient that there exist an increasing sequence (qn) of posi-

tive step functions such that

sup 9 < lima (x) if x e E.n n n n


In this statement one can replace "step functions" by "compact-

ly supported continuous functions".

0.5 CHANGE OF VARIABLES

Let V be an open set of ]R and u a diffeomorphism of V onto

u(V), that is to say a bijection of V onto u(V) such that u and

u-1 are continuously differentiable. The JACOBIAN of u is denoted

J(u).

PROPOSITION: The formula

x))IJ(x)ldxf(u(J

f(x)dx = Ju(V) V

is valid in each of the following two cases:

(1): f is a measurable positive arithmetic function on u(V);

(2): f is an integrable function over V.

Amongst other things the validity of the formula means that

fou is measurable (resp. (fou)IJI is integrable) on V.

Spherical Coordinates in ]R

e with the half-hyperplane defined by x1 < 0, x2 = 0 removed,

possesses a proper parametric representation:

x1 = rsinen_2 sin82sino1coscp,

x2 = rsinOn_2 sinO2sin81sin9,

= rsin sin8 cosx n_2 2 1,3

x4 = rsin4n_2 coso2,

xn = rcos8n_2,

OF THE COURSE 13

where r > 0, 0 < of < it, ITI < R. The formula for the change of

variables in this case comes down to replacing the xi by their

expression as a function of r, the oils, and cp, and dx = dx1dx2...

dxn by

rn-lsinn-ton-2... sin2o2sinoldrdSn-2...de2doldq.

Let us also recall that the volume of the ball x2 + +X2

<, 1n

is

n/2v_ n

n r 2 + 11

0.6 THE LP SPACES

In what follows we make the convention of setting 1/co = 0,

a.- = 0 or - according as a = 0 or a > 0, and coo = m if a > 0.

HOLDER'S INEQUALITY: Let 1 < p,q < - be such that

p1 +q =1,

and let f and g be measurable positive arithmetic functions on

31n; then:

Jfg `[JfPJ

l/p[Jgq/

1/q

MINKOWSKI'S INEQUALITY: Let 1 S p < c, and let f and g be measur-

able positive arithmetic functions on ]R", then

If (f + g)PJ 1/p <

[JfPJ1/p +

[JgPJ'/P.

When p = q = 2, Holder's Inequality is known as the (CAUCHY-


SCHWARZ INEQUALITY. For 0 < p < 1 one still has an inequality

which is obtained by replacing 4 with : in Minkowski's Inequality

(it is sometimes called MINKOWSKI'S SECOND INEQUALITY).

In order to have equality in Hblder's Inequality it is neces-

sary and sufficient that there exist a 3 0 such that fP = agq al-

most everywhere. For Minkowski's Inequality when 1 < p < -, equal-

ity is only obtained if f = ag almost everywhere.

NOTATION: For every measurable function f on IItn one sets

where 0<p<-.llfll = (Jlfll1/p

Also, there exists M e- such that lfl 4 M almost everywhere,

and such that this does not hold for any M' < M. The element M

is called the ESSENTIAL SUPREMUM Of lfl and one defines

lifilm = ess sup If I .

DEFINITION: With the preceding notations, for 0 < p << - one de-

fines the following functional spaces:

Lp(]Rn) { f: f is measurable on IItn; ll f llp < -).

These spaces are vector subspaces of the space of functions

on] .When 1 : p the mapping f i+ llflip is a semi-norm on LP (where

we write LP for LP(]R )); the kernel of this semi-norm coincides

with the vector subspace N of functions zero almost everywhere, so

that LP/N is canonically provided with a norm. In this work we

shall make not distinction between Lp and LP/N, which amounts to

identifying two functions that are equal almost everywhere. Let

us point out that certain authors denote by XP that which we de-

note by LP, and they keep the latter notation for .P/N. Having

taken into account the convention we have just indicated, in the

OF THE COURSE 15

remainder we shall consider f ' If 11 as a norm on LP (1,<p<,-).

THEOREM: Let 1 5 p 5 00 and let (un) be a sequence of elements of

LP such that

G iiunilp<n

Then for almost all x the series

u(x) = I u(x)n

is absolutely convergent; the function u thus defined almost

everywhere belongs to LP and one has

in the sense of convergence in the norm of L.

COROLLARY: For 1 < p 5 00 the spaces LP are complete. Furthermore,

if fn-> f in LP there exists a sub-sequence fn

.

such that fn

-r f

almost everywhere.

It will be noted that if fn -> f in L , then fn -> f almost every-

where. This property may fail if fn - f in LP, 1 p < 00.

When f e LP, g e Lq, 1 : p,q <

p

+

Q

= 1, the function fg is

integrable and

Jfgl 6 IIfIIpIIgIIq (Holder's Inequality).

The spaces LP (1 S p 5 00) are BANACH SPACES; L2 is even a Hil-

bert space if the scalar product is defined by

2(fig)= Jf0 f,g e L.


For every measurable set E of Rn one similarly defines the

spaces Lp(E) by everywhere replacing the integrals taken over Rn

by integrals over E. The space Lp(E) is identified with the

closed vector subspace of Lp(IZn) formed of the functions that

vanish outside E. When meas(E) < W one has the inclusions

Lq(E) C Lp(E) if 0 < p < q < -.

DENSITY THEOREM NO. 1: Let E be a measurable set of Rn; then the

characteristic functions of the measurable sets A C E are total

forLp(E) for 1SpSW.

DENSITY THEOREM NO. 2: Let V be an open set of Rn, then the char-

acteristic functions of the rectangles P such that P C V are total

in LP(V) for 1 4 p < The set of continuous functions compactly

supported in V are dense in Lp(V), 1 5 p <

Let us recall that a set A of a metric space E is said to be

DENSE in E if A = E; if E is a normed space, A is said to be TOTAL

in E if the linear combinations of elements of A are dense in E.

0.7 CONVOLUTION

DEFINITION: Two measurable functions f,g on Rn are said to be

CONVOLVAELE if the function y H f(x - y)g(y) is integrable for

almost all x; in this latter case one can define almost every-

Where the CONVOLUTION PRODUCT OF TWO FUNCTIONS f and g by

(f*g)(x) = Jf(x - y)g(y)dy.

We have:

f*g = g*f.

If f is convolvable with g a n d h, then it is with Ag + uh (A,1,

e ¢), and

OF THE COURSE

f*(Ag + uh) = A(f*g) '+ u(f*h).

17

If IfI 1, fl, IgI 1, gl, f and g are convolvable whenever fl and

5l are; from this it follows that if f = u + iv, g = r7+ is (u,v,

r,s real), f and g are convolvable if and only if each of the

functions u+,u_,v+,v_ is convolvable with each of the functions

r+,r_,s+,s_.

Whenever f and g are convolvable, and are zero respectively

outside measurable sets A and B, f*g is zero outside A + B. In

particular, if f and g are compactly supported, f*g is also com-

pactly supported.

In addition to the spaces Lp (i.e., Lp(]R )) introduced in the

preceding Section it is useful to define the following functional

spaces:

DEFINITION: If 1 < p < m we define LPoc to be the vector space of

measurable functions f such that for every compact set K of Rn one

has

HAIp,k =[JKIf1i'/p

<_-

The functions belonging to LPoc are said to be LOCALLY p-INTEGRABLE.

If p Lloc is the space of LOCALLY BOUNDED MEASURABLE FUNCTIONS,

that is to say that for every compact set K one has

Ilfi, K = ess sup If (X) I <

xeK

One can be satisfied with looking at compact sets of the type

KN = {x:Ixl <, N}, where N 3 1 is an integer and x -> IxI a norm on

RR, We then set

IIfIIp,K =n

IIfIIp,N,

and the formula


W IIf - gIIP,Nd(f,g) =

12-

N=11 +

If - gIIp,N

defines a metric on LPoc (with the usual condition of identifying

two functions; which are equal almost everywhere). This metric is

compatible with the vector space structure on LPoc, fi ; f if and

only if If - fiIIP,K 0 for every compact set K, and LPoc is com-

plete in this metric.

DEFINITION: The space of p-INTEGRABLE COMPACTLY SUPPORTED MEASUR-

ABLE FUNCTIONS is written LP for 1 < p 4 it is a vector sub-

space of LP. We say that fi - f in LP if fi I fin LP and if,

furthermore, there exists A > 0 such that for all i one has fi(x)

= 0 whenever IxI > A. This latter condition is expressed by say-

ing that the fi have their SUPPORT contained in a fixed compact

set.

DEFINITION: The SPACE OF k-FOLD CONTINUOUSLY DIFFERENTIABLE FUNC-

TIONS on ]Rn

is written Ek (0 s k s co). (For k = 0, E0 is defined

to be the space of continuous functions on 3zn; in this case we

may also write C instead of E0).

For every n-tuple s = (s1,...,sn) of integers greater than or

equal to zero one sets

Isl = s1 + .. + sn,

and we define

Ds =D I S I,ff

axs1 i...axnn '

feEk, IsI << k.

If 0 S k < -, for every integer N 3 1 and f e Ek we set:

pN(f) = sup{IDsf(x)I:Isj : k,lxl 4 N),

OF THE COURSE

and ifk= -,

pN(f) = sup{IDsf(x)I:Isj < N,jxj 4< N}.

With these notations the formula

19

W -N pN(f - g)

d(f,g) =N11

2- 1 +pN

f -g

defines a metric on Ek compatible with the vector space structure,

and for which Ek is complete. Furthermore, fi - f in Ek if for

every s such that Isl < k one has limDsf. = Dsf uniformly on every

compact set of ]R' .

DEFINITION: The vector space C is the vector SPACE OF BOUNDED

CONTINUOUS FUNCTIONS on ]R provided with the NORM

IIfL = suplf(x)I

It is a Banach space.

DEFINITION: The closed vector subspace of CW formed of UNIFORMLY

CONTINUOUS BOUNDED FUNCTIONS on ]R' is denoted UC`°.

DEFINITION: The space of CONTINUOUS FUNCTIONS of ]R" WHICH TEND TO

ZERO AT INFINITY is denoted CO. It is a closed vector subspace

ofUC .

DEFINITION: The vector subspace of Ek consisting of COMPACTLY

SUPPORTED k-FOLD CONTINUOUSLY DIFFERENTIABLE FUNCTIONS is written

Dk (0 5 k co). If k = 0 the,space D0, that is to say the space

of compactly supported continuous functions, is also written K.

We shall say that fi -> f in Dk if for all s (Isl s k) one has

limDsfi = Dsf uniformly on ]n and if the fi have their supporti

contained in a fixed compact set.


If E and F are two of the spaces that have just been defined

we shall write E - F if E C F and if fi - f in E implies that

fi -> f in F. It is clear that E - F and F -> G implies E -* G.We have the following diagram, where 0 < k < - and 1 < p <

E' --a Fk -p C - + L -' LPloc - Lloc beT

C

tUCW LW Lp L1

t

D"->Dk--* K ->LWc c c

It will be noted that D -> E for any space E from the table

above, and also E -> Lloc for every E; note also that the Lp spaces

are not mutually comparable.

NOTATION: For every function f and every a e Rn, we define the

TRANSLATION OF f BY a by

f.(x) = f(x - a).

Sometimes the notation Taf is used to designate fa. If f and g

are convolvable, then

(f*g)a = fa,:g = ff:ga.

DEFINITION: If E is one of the function spaces defined above, one

says that E is INVARIANT UNDER TRANSLATION if f e E implies that

fa e E for every a eRn. Furthermore, if ai -> a implies that f,'. -fI

in E, one says that the TRANSLATIONS OPERATE CONTINUOUSLY ON E.

OF THE COURSE 21

THEOREM: (1): The spaces LP,LlOC,LC (1 5 p Cm,UC,

D

CO,Ek and

k (0 5 k are invariant under translation;

(ii): The translations operate continuously on all these spaces,

except upon LW,L ,L and Cam.loc c

NOTATION-DEFINITION: If F,GH are three of the function spaces

defined above, the notation F::G C H expresses that if f e F, g e G,then f and g are convolvable and f*g e H, Furthermore, if fixgi -)-

fs:g in H whenever fi + f in F and gi -*-g in G, one writes F::G C H(continuously). In the case where F = G = H = A it is said that

A is a CONVOLUTION ALGEBRA. Lastly, if A is a convolution algebra

and A*E C E, one says that A OPERATES IN E; A is said to OPERATE

CONTINUOUSLY IN E if A^E C E continuously.

THEOREM: (i): L1*L C UCW (continuously); furthermore,

IIffgII <

(ii): Lp*Lq C C0 (continiously) if 1 < p,q < -, p + q = 1; fur-

thermore,

IIf*gL s IIfIIpIIgIIq;

(iii): L1s:Lp C LP (continuously) if 1 , p , m; furthermore,

11f* g11 , IIfII1IIgIIp.

In particular, L1 is a convolution algebra

(iv): L1operates continuously in Lp (1 4 p ,

oo),Cm,UC

and CO;

(v). Dk*Lloc C Ek (continuously), 0 < k

(vi): L1 it a convolution algebra;

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