A Course of Mathematical Analysis Volume II [2]

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S. M. NIKOLSKY

A Course of Mathematical Analysis Volume

2 MIR PUBLISHERS MOSCOW

Academician S.M NIKOLSKY a State Prize winner, the author of more than 130 scienti� c papers and several monographs including Quadrature Formulas, Approximation of Functions of Several Variables and Embedding Theorems and Integral Representation of Functions and Embedding Theorems (co-author). He contributed many fundamental results to the theory of approximation of functions, variational methods for solving boundary-value problems and functional analysis. For his monograph Approximation of Functions of Several Variables and Embedding Theorems he was awarded the Chebyshev Prize of the USSR Academy of Sciences.

A Course of Mathematical Analysis

C. M. HHKOJIbCKHft

KYPC MATEMATMM ECKOrO AHAJ1H3 A TOM 2

H3flATEJILCrBO «HAYKA» MOCKBA

S . M . N IK O L S K Y Member, USSR Academy o f Sciences

A Course of Mathematical Analysis i

Volume 2

Translatedfrom the Russian by V. M . VOLOSOV, D. Sc.

M IR PUBLISHERS MOSCOW

First published 1977 Second printing 1981 Third printing 1985 Fourth printing 1987

TO THE READER M ir Publishers would be grateful for your comments on the content, translation and design of this book. We would also be pleased to receive any other sugges­ tions you may wish to make. Our address is: USSR, 129820, Moscow 1-110, GSP Pervy Rizhsky Pereulok, 2 M IR PUBLISHERS

Ha QH2AUUCKOM H3HK€ © H w ien b c iB o «HayKa», 1975 © English translation, M ir Publishers, 1977

Contents Chapter 12. § 12.1. § 12.2. § 12.3. § 12.4. § 12.5. § 12.6. § 12.7. § 12.8. § § § § § § § § § § § § § § §

Multiple Integrals .................................................................................. Introduction .......................................................................................... Jordan Squarable Sets .......................................................................... Some Important Examples of Squarable S e ts...................................... One More Test for Measurability of a Set. Area in Polar Coordinates. Jordan Measurable Three-dimensional and /i-dimensional Sets........ The Notion of Multiple In teg ral........................................................... Upper and Lower Integral Sums. Key Theorem ................................ Integrability of a Continuous Function on a Measurable Closed Set. Some Other Integrability Conditions ................................................ 12.9. Set of Lebesgue Measure Zero ................................. 12.10. Proof of Lebesgue’s Theorem. Connection Between Integrability and Boundedness of a Function ..............: ................................................ 12.11. Properties of Multiple Integrals .......................................................... 12.12. Reduction of Multiple Integral to Iterated Integral ....................... 12.13. Continuity of Integral Dependent on Param eter................................ 12.14. Geometrical Interpretation of the Sign of a Determinant ................ 12.15. Change of Variables in Multiple Integral. Simplest C a se ................. 12.16. Change of Variables in Multiple Integral. General Case ................. 12.17. Proof of Lemma 1,§ 12.16 12.18. Double Integral in Polar Coordinates.................................................. 12.19. Triple Integral in Spherical Coordinates .......................................... 12.20. General Properties of Continuous Operators ................................... 12.21. More on Change of Variables in Multiple Integral ......................... 12.22. Improper Integral with Singularities on the Boundary of the Domain of Integration. Change of V ariables.................................................... 12.23. Surface Area .......................................................................................

Chapter 13. Scalar and Vector Fields. Differentiation and Integration of Integral with Respect to Parameter. Improper Integrals ............................... § 13.1. Line Integral of the First Type .......................................................... § 13.2. Line Integral of the Second Type ....................................................... § 13.3. Potential of a Vector Field .................................................................. § 13.4. Orientation of a Domain in the Plane ............................................... § 13.5. Green’s Formula. Computing Area with the Aid of Line Integral .. § 13.6. Surface Integral of the First Type ....................................................... § 13.7. Orientation of a Surface........................................................................ § 13.8. Integral over an Oriented Domain in the P la n e ................................. § 13.9. Flux of a Vector Through an Oriented Surface ................................ § 13.10. Divergence. Gauss-Ostrogradsky Theorem ....................................... § 13.11. Rotation of a Vector. Stokes’ T heorem ............................................... 5

9 9 11 17 19 20 24 27 32 34 35 38 41 48 51 54 56 59 63 65 67 68 71 73 80 80 81 83 91 92 96 98 102 104 107 114

CONTENTS

6 § § § § §

13.12. Differentiation of Integral with Respect to Parameter ..................... 13.13. Improper Integrals ............................................................................... 13.14. Uniform Convergence of Improper Integrals ................................... 13.15. Uniformly Convergent Integral over Unbounded D o m ain ................ 13.16. Uniformly Convergent Improper Integral with Variable Singularity..

118 121 128 135 140

Chapter 14. § 14.1. § 14.2. § 14.3. § 14.4. § 14.5. § 14.6. § 14.7. §14.8. § 14.9.

Normed Linear Spaces. Orthogonal Systems...................................... Space C of Continuous Functions ................................... Spaces L \ Lp and l9 ............................................................................ Spaces L't and Lt ................................................................................ Approximation with Finite Functions .................................................* Linear Spaces. Fundamentals of the Theory of Normed Linear Spaces Orthogonal Systems in Space with Scalar Product .......................... Orthogonalization Process ................................................................. Properties of Spaces ££(£) and L2(£ ) ................................................ Complete Systems of Functions in the Spaces C, L ’t and U (L2, L)

147 147 149 154 156 163 170 181 185 187

Chapter 15. § 15.1. § 152. § 15.3. § 15.4. § 15.5.

Fourier Series. Approximation of Functions with Polynomials......... Preliminaries ....................................................................................... Dirichlet’s Sum ..................................................................................... Formulas for the Remainder of Fourier’s Series................................... ■ Oscillation Lemmas ........................................................................... Test for Convergence of Fourier Series. Completeness of Trigono­ metric System of Functions ............................................................. Complex Form of Fourier Series ........................................................ Differentiation and Integration of Fourier Series ........................... Estimating the Remainder of Fourier’s Series ................................... Gibbs’ Phenomenon............................................................................... Fej£r’s:S um s........................................................................................... Elements of the Theory of Fourier Series for Functions of Several V ariables................................................................................................ Algebraic Polynomials. Chebyshev’s Polynomials ........................... Weierstrass’ Theorem ......................................................................... Legendre's Polynomials .......................................................................

188 188 195 197 199

§ § § § § §

15.6. 15.7. 15.8. 15.9. 15.10. 15.11.

§ 15.12. § 15.13. § 15.14.

Chapter 16. Fourier Integral. Generalized Functions § § § § § § § § § § §

203 211 213 216 217 221 225 235 236 237

...........................................

240

16.1. Notion of Fourier Integral ............................................................... 16.2. Lemma on Change of Order of Integration ...................................... 16.3. Convergence of Fourier’s Single Integral .......................................... 16.4. Fourier Transform and Its Inverse. Iterated Fourier Integral. Fourier Cosine and Sine Transforms.................................................................. 16.5. Differentiation and Fourier Transformation ...................................... 16.6. Space S ................................................................................................ 16.7. Space S ' of Generalized F unctions..................................................... 16.8. Many-dimensional Fourier Integrals and Generalized Functions__ 16.9. Finite Step Functions. Approximation in the Mean Square............... 16.10. Plancherel's Theorem. Estimating Speed of Convergence of Fourier’s Integral....................... „.......................................................................... 16.11. Generalized Periodic Functions ..........................................................

240 243 245

Chapter 17. § 17.1. § 17.2. § 17.3. § 17.4.

Differentiable Manifolds and Differential F o rm s............................... Differentiable Manifolds ..................................................................... Boundary of a Differentiable Manifold and Its O rientation............. Differential F o rm s................................................................................ Stokes’ Theorem .............................

247 249 250 255 265 273 278 283 289 289 299 310 220

CONTENTS

7

Chapter 18. Supplementary Topics ......................................................................... § 18.1. Generalized Minkowski’s Inequality ................................................. § 18.2. Sobolev’s Regularization of Function ............................................... § 18.3. Convolution ....................................................................................... § 18.4. Partition of U nity.................................................................................

326 326 329 333 335

Chapter 19. Lebesgue Integral ................................................................................. § 19.1. Lebesgue M easure............................................................................... § 19.2. Measurable Functions ........................................................................ § 19.3. Lebesgue integral ............................................................................... § 19.4. Lebesgue Integral on Unbounded Set ............................................... § 19.5. Sobolev’s Generalized Derivative ..................................................... § 19.6. Space D ' of Generalized Functions ................................................... § 19.7. Incompleteness of Space L*p ................................................................ § 19.8. Generalization of Jordan M easu re.................................................... § 19.9. Riemann-Stieltjes Integral .................................................................. § 19.10. Stieltjes In teg ral................................................................................... § 19.11. Generalization of Lebesgue Integral ............................................... § 19.12 Lebesgue-Stieltjes Integral.................................................................... § 19.13. Extension of Functions. Weierstrass’ Theorem ............................... Name Index ............................................................................................................... Subject Index . ..............................................................................................................

338 338 348 355 388 392 404 407 408 414 415 423 424 433 437 438

The major part of this two-volume textbook stems from the course in mathematical analysis given by the author for many years at the Moscow Physico-technical Institute. The first volume consisting of eleven chapters includes an introduction (Chapter 1) which treats of fundamental notions of mathematical analysis using an intuitive concept of a limit. With the aid o f visual interpretation and some considerations of a physical character it establishes the relationship between the derivative and the integral and gives some elements of differen­ tiation and integration techniques necessary to those readers who are simultaneously studying physics. The notion of a real number is interpreted in the first volume (Chapter 2) on the basis of its representation as an infinite deci­ mal. Chapters 3-11 contain the following topics: Limit of Se­ quence, Limit of Function, Functions of One Variable, Func­ tions of Several Variables, Indefinite Integral, Definite Integral, Some Applications of Integrals, Series.

CHAPTER 12

Multiple Integrals

§ 12.1. Introduction Let us consider a continuous surface, lying in the three-dimensional space with rectangular coordinates (x, y 9 z )9 which is determined by an equation

z

= /( ® = /(*»

y)

(Q = (x, y)

e Q)

where Q is a bounded (two-dimensional) set possessing area (two-dimen­ sional measure*). For instance, Q can be a circle, a rectangle, an ellipse, etc. We shall suppose that the function /( x , y) is positive. Let us state the following problem: it is required to find the volume of the solid bounded above by the given surface and below by the plane z = 0, its lateral boun­ dary being the cylindrical surface with generators parallel to the z-axis and passing through the boundary curve y o f the set Q. To determine the sought-for volume we resort to the following natural procedure. The set Q is divided into a finite number N of parts (subdomains)

Q\9 . . Qy

(1)

any two of which either do not intersect or intersect only along some parts of their boundaries. Let these subdomains be such that they possess areas (two-dimensional measures) which we shall denote as m£2l9 . . . , thQn respectively. Let us introduce the notion of the diameter of a set: if A is a set in the plane its diameter d(A) is defined as

d(A )=

sup

\P '-P " \

where the supremum is taken over all the pairs of points P \ P" belonging to A. Now we choose an arbitrary point Qj = (ly, rjj) ( j = 1, . . N) in each part Qj and form the sum

VN = Z fm m Q j i * See § 12.2.

(2)

A COURSE OF MATHEMATICAL ANALYSIS

10

which can be regarded as an approximation to the sought-for volume V. We can naturally suppose that the smaller the diameters d(Qj) of the subdomains Qj are, the higher is the accuracy of the approximation V ^ Therefore the volume V of the solid in question can be defined as the limit

V=

lim

Y nQ jim Q j

(3)

max d(Qj)-*Q

to which sum (2) tends when the maximum diameter of the subdomains of partitions (1) are made to tend to zero provided that this limit exists and is independent of the way in which the sequence of partitions (1) is chosen. Now we can abstract from the problem of finding the volume of a solid and regard expression (3) as the result of an operation performed on the given function / defined in Q. It is called the Riemann double integral o f the function f over the domain Q and is denoted

V=

Um

Z fm r n Q j = f f /( x , y) dx dy = f/< 0 ) dQ = f fd Q

max d(Qj)-+0 Jsa^

J

J

Let us consider a problem leading to the notion o f the triple integral. Suppose that there is a physical body occupying a domain (set) Q in the three-dimensional space with rectangular coordinates (x, y, z) and that the mass of the body is distributed (nonuniformly, in the general case) over Q with volume density p (x, y 9 z) = p(0 ) (Q = (x, y 9 z) € &)• It is required to determine the total mass of the body Q. To solve this problem it is natural to partition Q into N parts fli, .. Qn whose volumes (three-dimensional measures) are mQu ...» mQN (on condition that these volumes exist), to choose an arbitrary point Qj = = (xJyyj>zj) € Q U = . . . , N) in each of the parts and to define the soughtfor mass as the. limit

M =

lim

£ yiQfirnQj

(4)

max d{O j)-^0

Expression (4) can again be regarded as the result of an operation per­ formed on the function [i defined in the three-dimensional set Q. It is called the Riemann triple integral o f f on Q and is denoted as

M =

lim max d(Qj)-+ 0

£ n(Qj)mQj = j (i(Q) dQ = j j j fi(x, y, z) dx dy dz

Q Q The Riemann nfold multiple integral is defined in the same way. We shall see that the theory of (Riemann) multiple integration which includes existence theorems and theorems on the additive properties of the integral can be presented for the /i-dimensional case in exactly the same manner as in the case of dimension 1. However, the theory of multiple integrals involves some specific difficulties which were not encountered in the theory of one-fold integration.

MULTIPLE INTEGRALS

11

The matter is that the (Riemann) one-fold integral was defined for an extremely simple set, namely, for a closed interval [a, b] which was parti­ tioned into parts which were also closed intervals. Therefore we had no diffi­ culties in defining the lengths (one-dimensional measures) of the intervals. But in the case of a double integral or, generally, n-fold integral, the domain of integration Q can be split into parts with curvilinear boundaries, which makes it necessary to define the notion of the area or, generally, of the H-dimensional measure of such a part. A similar question would also appear in the case n = 1 if we defined the one-fold Riemann integral for a set of a more complex structure than that of a closed interval. In this connection we must state a strict definition of the notion of measure of a set and investigate the properties of the measure. Therefore we begin this chapter with the theory of the Jordan* measure closely related to the theory of the Riemann integral. This theory forms the basis for the repre­ sentation of the theory of the Riemann multiple integral. The latter theory provides an important method for evaluation of /i-fold multiple integrals by reducing them to the so-called iterated (repeated) integrals involving n one-fold integrations with respect to each of the variables; in many impor­ tant cases this procedure admits of the application of the Newton-Leibniz theorem established for one-fold integrals. § 12.2. Jordan Squarable Sets Let us consider the plane R = R 2 with a definitely chosen rectangular coordinates (x, y); this coordinate system will also be denoted by the same letter R. If some other coordinate system (I, rj) is taken in the same plane we shall denote the plane (and the new coordinate system) by R'. A rectangle A in the plane R will be regarded as the simplest set. It can be defined analytically by assuming that there is a system of rectangular coordinates Rr in which A is representable as a set of points (I, rj) satisfying inequalities of the form fli

I < a2,

(1)

where au a2, &i and b2 are some numbers such that a± < a2 and bi < b2. The coordinate system R' possesses the property that the sides of A are parallel to its coordinate axes. To stress that the sides of A are parallel to the coordinate axes of the system R' we shall write A = Ar*. The rectangles of the type of A are understood here as closed sets (closed rectangles in­ cluding their boundaries). Now we define the notion of an elementary figure o: a set o n (G) c

G c &n (G)

where N and N ' are arbitrary natural numbers. It follows that there exist the finite limits miG = lim | ^ | and meG = lim | j. / Now we can take in the co­ i i 9 // 1 ordinate system R a network f * ( to 1 t < a ; then the set o " —o' en­ / z / s tirely contains every square of J f the network which covers at I least one boundary point of o. Fig. 12.2 Therefore the sum of the areas of all the squares of S n covering the boundary of crdoes not exceed |cr" —cr'| = \o"\ — | Gi, o r z> G2, \ o 'r \ < y and \oR | < . Therefore for the figure oR = o'r+Or we have 4

3 6 i+ G j

and

|o*|

|n(E)\ z>\A\ > 0 and meEz> \A\ > 0 . Hence, rriiE -< meE, which means that E is a nonsquarable set. Let us prove the additive property of the Jordan measure. Theorem 2. I f two sets Gi and G2 are Jordan measurable and if their com­ mon points, provided there are such, only belong to their boundaries, their union

(which is measurable by Theorem 1) has a measure equal to the sum o f the measures o f the sets Gi and G2: ih(G i + G 2) = mGi+/wG2 (7) Proof Let us choose an arbitrary e > 0 and take some figures o[ = o'itR, Gi = a”,r 9o2 = o2tR and o2 = o2tR such that / tt / tt oi c Gi c (Ti, o2 c G2 c o2 ♦

mGx—e < |o i| < \o± | < m G i+ e and mG2—e < | £0; for such fc’s the subsets r belong to a', that is

m£2k =

adjoining

■< l^ l < e

j Theorem 4. The definite integral o f a boundedfunction f on a measurable set £2 can be defined as the limit = \fd x Q

lim dQM^° J

fo r a concrete sequence o f partitions (9) with bQi -►0 where the integral sums only extend over the subsets £2} not adjoining jT. For, by the foregoing theorem, the parts of the integral sum corresponding to these £2f s which adjoin T can be estimated by the inequality (K > i/(* )i)

- 0,

§ 12.8. Integrability of a Continuous Function on a Measurable Closed Set. Some Other Integrability Conditions Theorem 1. A function f(P ) continuous on a Jordan measurable closed set

£2 is Riemann integrable on £2. Proof Since the set £2 is measurable it is bounded. Besides, it is closed, and therefore the function / is uniformly continuous on £2. This means that for any e > 0 there is d > 0 such that if F 9 P” £ £2 and \P"—P'\ < b then |/( P " ) - /( P ') I < s. N Let q be an arbitrary partition £2 = Y fij of the set £2 into measurable i parts £2j with diameters d(£2j) < 6, and let, as usual, Mj = sup f( x ) and *€ Q$ mj = inf /(x ). We have *€ % M j-m j = sup |f i n - f i P y ^ e P'.r'tOj because, by the hypothesis, the distance between any points F , P " 6 £2j does not exceed b. Consequently,

SQ-~Se =

X

(Mj-mj)m£2 j
0 can be made arbitrarily small, the key theorem shows that the integral of / on £2 does in fact exist. Fheorem 2. A function f defined and bounded on a measurable closed set £2 and continuous on £2 everywhere except possibly at points forming a subset A 0 be an arbitrary posi­ tive number and let a z> A be an open (without boundary) figure such that | a | Then Q —o is a measurable closed set on which / is continuous and, consequently, integrable. Let us take a partition g' of the set £2—o of the form £2—a = . . . +£2N such that So and then construct the partition g of the form £2 = Q ± + . . . +£2n+oQ of the set Q. On putting M = sup /( x ) and m = inf /( x ) we write as€o0 as€aO = { S j - S f ) + (M —m)MoQ) < e+ 2 £ e = q (tf= H /(* )l, x g f i ) where, 77 can be made arbitrarily small. Therefore, according to property (ii) mentioned in the key theorem, the function/is integrable on £2. It turns out that Theorem 2 can be generalized to the case when the set A is of Lebesgue measure zero (instead of Jordan measure zero; see§ 12.9). This generalization provides a necessary and sufficient condition for the Jordan integrability of a bounded function on a measurable closed set be­ cause there holds the following theorem: Theorem (H. L. Lebesgue). A functionf(x) defined and bounded on a Jordan measurable closed set £2 is Riemarm integrable on £2 if and only if thefunction

f is continuous on £2 except possibly on a subset o f £2 having Lebesgue measure zero. Example. Let us take the function y(x, y) = s i n — and consider it as dexy fined in the half-open rectangle A’ = (0 x, y < n/2). To apply Theorem 2 to this function we can argue in the following way. Let us extend ip to the closed interval 0 x n jl of the x-axis and to the closed interval O ^ y ^ n jl of the y-axis in an arbitrary manner so that the new values of the function are uniformly bounded. The extended function ip is then defined on the closed rectangle A = jj' and is bounded on A and continuous on A everywhere except on the set of Jordan two-dimensional measure zero (consisting of these two closed intervals). Consequently, by Theorem~2, the integral

j j V>(x, y)d xd y = j j y(x, y)d xd y A

A

exists (see§ 12.11, Theorem 1 and the corollary of this theorem).

34

A COURSE OF MATHEMATICAL ANALYSIS

§ 12.9. Set of Lebesgue Measure Zero* An arbitrary open rectangular parallelepiped (rectangle)

A -{ a j < Xj «= bj\ j = 1, . . n) in the w-dimensional space R = R„ will also be referred to as an interval in R. The volume (/i-dimensional measure) of A is equal to M l = f [ (bj-aj) i because a closed rectangle differs from the corresponding open rectangle by a set of (n-dimensional) measure zero. We say that a set E is o f Lebesgue measure zero if for any e > 0 there is a finite or countable number of intervals A \ A29 . . . covering E (E c the sum of whose volumes is less than e: £ \Ak | < e. Lemma 1. The sum o f a finite or countable number o f sets E l9F 2, . . . each

o f which has Lebesgue measure zero is in its turn a set o f Lebesgue measure zero. For, if e > 0 is an arbitrary positive number we can construct the fol­ lowing coverings of the given sets: each Ek is covered by a countable (or finite) System of intervals A) ^Ek c such that £ \Ak\ *< ej2k (k = 1, 2 , . . . ) . Since the intervals Af ( j 9 k = 1,2, . . . ) can be renumbered as A l9 A 2, . . . and since their union covers E = £ E k and the sum of their volumes is less than e = £(e/2k) we see that, by the arbitrariness of e9 the set E

k is of Lebesgue measure zero. If a set E has a Jordan measure zero this means that for any e > 0 it can be covered by a finite number of intervals with total volume less than e9 and consequently such a set J? is also of Lebesgue measure zero. This shows that it is permissible to use one and the same notation (i.e. to write mE = 0) for the Jordan and the Lebesgue measure when the Jordan measure exists and is equal to zero. However, here we have only discussed a very special case of the Lebesgue measure, namely, the case when it is equal to zero. It should be stressed that a set can be of Lebesgue measure zero and at the same time be Jordan nonmeasurable. For example, the set of rational points contained in the closed interval [0, 1] has Lebesgue measure zero (since it is countable). But it has no (one-dimensional) Jordan measure be­ cause its exterior Jordan content is equal to 1 while the interior Jordan content is equal to zero. We also note that if a set F is bounded and closed and has Lebesgue mea­ sure zero then, given any e 0, there is a countable system of intervals covering F whose total volume is less than e. By the Heine-Borel lemma. *The subject matter of this section is a part of the material presented in full in§ 19.1.

MULTIPLE INTEGRALS

35

since the set F is bounded and closed, this system contains a finite subsystem covering F, and the sum of the volumes of the intervals in this subsystem is less than s. Therefore F is Jordan measurable and has the Jordan measure equal to zero. We have thus proved the following lemma. Lemma 2. A bounded closed set o f Lebesgue measure zero is Jordan mea­ surable and its Jordan measure is also equal to zero. § 12.10. Proof of Lebesgue’s Theorem. Connection Between Integrability and Boundedness of a Function In § 7.10 we stated the definition of the oscillation a>(x) ( co( x ) > 0) of a function / , defined on a set Q, at a point x £ Q and showed (see § 7.10, Theorem 5) th a t/is continuous at x if and only if its oscillation at that point is equal to zero (co(x) = 0). Hence, the oscillation of a function at its point of discontinuity is sure to be positive. Let Ex denote the set of all points x € Q at which the oscillation of / is not less than A > 0 (a>(x) s* A). It is important to npte that if Q is a closed set then so is the set Ex for any A (see § 7.10, Theorem 6). Proof o f sufficiency of the condition of Lebesgue’s theorem. Let the func­ tion / b e bounded on a measurable closed set Q, that is

\f(x )\^ K

(x e fl)

(1)

and let the set E of its points of discontinuity be of Lebesgue measure zero

(E czQ). We shall suppose that mQ > 0 since if otherwise the assertion we have to prove becomes quite obvious. Let us take an arbitrary e > 0, and let A > 0 satisfy the inequality 4XmQ •< e. Since we have mE = 0 for the Lebesgue measure of F , the Lebesgue mea­ sure of Ex is also equal to zero for any A > 0 (mEx = 0). But the set Ex is bounded and closed, and therefore the Jordan measure of Ex is also equal to zero. Consequently, given any e > 0, there is a set G covering Ex which is a finite system of intervals with measure |G | < The set G is open and measurable and, since the set Q is bounded and closed it is representable as a union of two nonintersecting Jordan measurable sets: Q = Qx+Q" where = GQ9Q" = G9m£2i |G | < and Q" is a bounded closed set The oscillation of the function in question is less than A at each point of the set Q". Therefore, by Theorem 3, § 7.10, there is b > 0 such that |/( F ) — •‘-/(P ')l < 2A for any two points F , Pr 6 satisfying the condition IF—P ' | < 5. Let us partition Q" into parts of diameters less than the numV

AT

dct o : Q” =

These parts together with the set

defined above form

2

a partition q of the whole set Q. For this partition we obviously have the

36

A COURSE O F MATHEMATICAL ANALYSIS

inequalities

S q—S q = (M i—

Yj (My—mj)mQj< 2 « s2 tf-^ + 2 A £

< -+ 2 A m fi < y + - = «

Consequently, by the arbitrariness of £ and by the key theorem, the func­ tion / is integrable on Q. Proof o f necessity of the condition of Lebesgue’s theorem. Let the func­ tion / b e bounded on a measurable closed set Q and integrable on it. Then, according to the key theorem, for any e > 0 and A > 0 there exists a parti­ tion q of the set Q such that (see the explanations below) el > S q—S q = (My—mj)mQj > Y l (My—my)m Q j A£'m£?y (2) The sum extends only over those summands which correspond to the sets Qj each of which contains at least one interior point P of Ex- Such a point can be covered by a ball V&with centre at that point lying within Qj, and therefore we have the inequalities My—my => M j— co(P) A and M* = sup /(x ), m* = inf /(x ) On cancelling (2) by A we obtain the inequalities

e > %mQj = m & 'Q j) where £'£?y contains the points of i?* each of which is an interior point of one of the subsets Qj entering into the partition q. The remaining points of Ex not belonging to Y 'Q j CSLn onty on boundaries of the sets Q j(j = = 1, . . . , N) whose total measure is zero. Thus, for any A > 0 and e > 0 we have mEx that is mEx = 0. Now, since the set of all points of dis­ continuity o f/c a n be represented in the form

E = Y E lfk k=*l

and since mEuk = 0 (k = 1, 2, . . . ) we conclude that mE = 0. The necessity of the condition of the theorem has been proved. Let us state the following useful definition. We shall say that a set Q c R possesses property (A) if it is measurable and if for any £ > 0 there is a partition Q =

N

° f ® into measurable parts of positive measures (mQj >- 0)

i with diameters d(Qj) < £. An arbitrary measurable open set Q possesses property (A). For, if we take a rectangular network with cubes A of diameter less than £, it partitions Q into nonempty measurable parts Qj. Let a set Qj of this kind be a part o f A and let x° c Qj c Q; then there is a ball V& with centre at x° contained in

MULTIPLE INTEGRALS

37

Q9 and the intersection of F*o and A belongs to Qy V&A c Qj. Now it is readily seen that miy& A) > 0. O f course, together with the measurable open set Q9its closure Q also pos­ sesses property (A). A rectangle, a circle and an ellipse are all examples of two-dimensional sets possessing property (A). Examples of one-dimensional sets possessing property (A) are a closed interval, a finite system of closed intervals and an open interval; as examples of three-dimensional sets with property (A) we can take a ball, a cube, an elementary figure, an ellipsoid and a torus.

On the other hand, the plane set Q shown in Fig. 12.4 which consists of the circle a and of the closed interval [0,2] lying in the x-axis does not pos­ sess property (A). This set is measurable because its boundary consists of a finite number of smooth curvilinear segments. However, if, for instance, we take e < 1/2, it is readily seen that it is impossible to b reak u p ^ into measur­ able (in the two-dimensional sense) parts of positive measures with diame­ ters less than e. Theorem 1. A function f integrable on a set Q possessing property (A) is

bounded on Q. Indeed, let us take an arbitrary 5 > 0 and construct a partition q o f the set Q into parts Qj ( j = 1, . . . , JV) of positive measures so that dQ= max d(Qj) < d j Suppose that the fu n ctio n /is unbounded on Q; then it is unbounded on at least one of the sets Qj9 say on Qi. The corresponding integral sum can be represented as = f(pi)m Q 1+ £ f(pj)mQj 7=2 For the given q and fixed pj (j = 2, . . . , N) the sum £ f(p j) mQj is constant ,.

7=2

wnue the product f(pi)m Q i wherep 1 is a variable point belonging to Qi is

A COURSE OF MATHEMATICAL ANALYSIS

38

unbounded (because mQ\ > 0). Consequently, the integral sum SQis un­ bounded. This shows that SQcannot tend to a finite limit as dQ-+■ 0, and consequently the function / is nonintegrable on Q (cf. Theorem 1, § 9.2). For the sets possessing property (A) Lebesgue’s theorem can obviously be stated as follows. Theorem 2. Let Q be a closed set possessing property (A). Then afunction f

defined on Q is integration Q i f and only if it is bounded on Q and continuous on Q except possibly at the pointsforming a subset o fQ o f measure zero. c

§ 12.11. Properties of Multiple Integrals Theorem 1. I f a function f is bounded and integrable on a set Q = Q'-hQ”

where Q‘ and Q” are measurable sets which either do not intersect or intersect only along some parts o f their boundaries then f is also integrable on each o f the sets Q' and Q" and vice versa. In this case* j f d x = jfd x + j f d x Q O' Q"

(1)

To prove the theorem we take an arbitrary sequence of partitions g* {k = = 1,2, . . . ) of the set Q such that the boundaries of the sets Q' and Q" consist only of the boundary points of some of the subsets into which Q is partitioned. These partitions induce on Q' and Q" the corresponding parti­ tions Qk and g*\ The further course of the proof is exactly the same as that of the proof of Theorem 1 in§ 9.7 for the one-dimensional integral* the role of the closed intervals [a, c] and [c, b] being played by the sets Q' and Q”. Corollary. I f a bounded and integrable function f defined on Q is redefined on any subset E c Q o f Jordan measure zero so that the modifiedfunction f \ remains bounded on Qy then f i is integrable on £2 and jfd Q = jfid Q

Q Q Indeed, the set Q—E is measurable together with f2, and therefore/is in­ tegrable on Q—E and, besides, J fd Q = jfxdQ = E

0

E

* For an unbounded function/equality (1) may not hold. For instance, i f / = 0 on o’ (see Fig. 12.4) and/ = 1/ jcon the half-open interval (0,2] of the a^ax isth en JJ/d x d y =

a = j j f d x d y = 0. At the same time the integral j j f d x dy does not exist,

y

a+y

MULTIPLE INTEGRALS

C o n seq u en tly ,/i is integrable on

£2 and

J fd£2 + jfd Q = j f d Q

jfid £ 2 = j f i d Q + j f d Q ^ Q

Q -E

39

E

Q -E

E

Q

By virtue of this assertion, in the case when a function / is bounded on a measurable nonclosed set £2 and integrable on £2 it makes sense to write

\fd x = jfd x 2T

Q

although the function/ may not be defined onQ —Q. For, if/is not originally defined on £2 and is then extended to Q—Q in an.arbitrary way so that its values assumed on £2—£2 are uniformly bounded, the integrals of f over £2 and over Q coincide. Theorem 2. Iff(x ) and q>(x) are bounded integrable functions on £2 and c is

a constant then the functions (i)f(x)±q>(x )9 (ii) c/(x),

(iii) |/( x ) |,

(iv)f(x) d > 0) are integrable on £2. Be­ sides, J(/±9>)