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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book is devoted to integration, one of the two main operations in calculus.
In Part 1, the definition of the integral of a one-variable function is different (not essentially, but rather methodically) from traditional definitions of Riemann or Lebesgue integrals. Such an approach allows us, on the one hand, to quickly develop the practical skills of integration as well as, on the other hand, in Part 2, to pass naturally to the more general Lebesgue integral. Based on the latter, in Part 2, the author develops a theory of integration for functions of several variables. In Part 3, within the same methodological scheme, the author presents the elements of theory of integration in an abstract space equipped with a measure; we cannot do without this in functional analysis, probability theory, etc. The majority of chapters are complemented with problems, mostly of the theoretical type.
The book is mainly devoted to students of mathematics and related specialities. However, Part 1 can be successfully used by any student as a simple introduction to integration calculus.
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Seitenzahl: 239
Veröffentlichungsjahr: 2014
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Contents
Preface
Note for the Teacher or Who is better, Riemann or Lebesgue?
Notation
PART 1 Integration of One-Variable Functions
1 Functions without Second-kind Discontinuities
P.1. Problems
2 Indefinite Integral
P.2. Problems
3 Definite Integral
3.1. Introduction
P.3. Problems
4 Applications of the Integral
4.1. Area of a curvilinear trapezium
4.2. A general scheme for applying the integrals
4.3. Area of a surface of revolution
4.4. Area of curvilinear sector
4.5. Applications in mechanics
P.4. Problems
5 Other Definitions: Riemann and Stieltjes Integrals
5.1. Introduction
P.5. Problems
6 Improper Integrals
P.6. Problems
PART 2 Integration of Several-variable Functions
7 Additional Properties of Step Functions
7.1. The notion “almost everywhere”
P.7. Problems
8 Lebesgue Integral
8.1. Proof of the correctness of the definition of integral
8.2. Proof of the Beppo Levi theorem
8.3. Proof of the Fatou–Lebesgue theorem
P.8. Problems
9 Fubini and Change-of-Variables Theorems
P.9. Problems
10 Applications of Multiple Integrals
10.1. Calculation of the area of a plane figure
10.2. Calculation of the volume of a solid
10.3. Calculation of the area of a surface
10.4. Calculation of the mass of a body
10.5. The static moment and mass center of a body
11 Parameter-dependent Integrals
11.1. Introduction
11.2. Improper PDIs
P.11. Problems
PART 3 Measure and Integration in a Measure Space
12 Families of Sets
12.1. Introduction
P.12. Problems
13 Measure Spaces
P.13. Problems
14 Extension of Measure
P.14. Problems
15 Lebesgue-Stieltjes Measures on the Real Line and Distribution Functions
P.15. Problems
16 Measurable Mappings and Real Measurable Functions
P.16. Problems
17 Convergence Almost Everywhere and Convergence in Measure
P.17. Problems
18 Integral
P.18. Problems
19 Product of Two Measure Spaces
P.19. Problems
Bibliography
Index
To my beloved wife Eugenija
First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2014The rights of Vigirdas Mackevičius to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2014945514
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-769-0
Preface
This textbook is devoted to integration, an important part of calculus. In Part 1, the definition of the integral of a one-variable function is different (not essentially, but rather methodically) from traditional definitions of Riemann or Lebesgue integrals. Such an approach allows us, on the one hand, to quickly develop practical skills of integration and, on the other hand, later, in Part 2, to pass naturally to the more general Lebesgue integral. Based on the latter, in Part 2, we develop a theory of integration for functions of several variables. In Part 3, within the same methodological scheme, we present the elements of theory of integration in an abstract space equipped with a measure; we cannot do without this in functional analysis, probability theory, etc. The majority of the chapters are complemented with problems, mostly of the theoretical type.
Although the three parts of the book are methodically related to each other, they are somewhat independent. For example, any reader accustomed to the Riemann integral and wishing to get into the theory of the Lebesgue integral is encouraged to begin with Part 2. Those who feel they are lacking in the basics of general theory of measure and integration should open the book from Part 3.
The book is mainly devoted to students of mathematics and related specialities. However, Part 1 can be successfully used by any students as a simple introduction to integration calculus.
We use the double numbering of statements (theorems, propositions, etc.): the first number denotes the number of the chapter, and the second indicates the number of the statement within the chapter. Though a large number of books and papers have influenced the contents of this book, the short reference list includes only those directly used by the author.
The book is essentially a revised translation of the author’s book Integral and Measure (TEV, Vilnius, 1998) from Lithuanian. The author would like to thank Vilijandas Bagdonavičius and a large number of students of the Faculty of Mathematics and Informatics of Vilnius University. Thanks to them, the book contains significantly fewer misprints.
Vigirdas MACKEVIČIUSVilnius, July 2014
Note for the Teacher or Who is better, Riemann or Lebesgue?
Which integral, that of Riemann or Lebesgue, is preferable in teaching calculus to first-year students of mathematics? In discussions on this question, the opponents present serious arguments in favor of “their” integral and, as a rule, never change their opinion. The author of this textbook, although a supporter of the Lebesgue integral, proposes the third approach to teaching integration (first, for the first-year students). It is mentioned in some mathematical books; however, I did could not find it in textbooks for university students. In my opinion, the main advantages of the approach are the following: (1) a relatively simple definition of the integral of one-variable functions and the proof of its existence and main properties, (2) a natural passage to the Lebesgue integral (of multi-variable functions and in abstract measure spaces) when the students already have the basic skills of integration and its applications.
To comprehend the approach, let us recall, in a few words, the Daniel construction of the Lebesgue integral. First, the integral is defined for some class of “simple” functions, say step functions, and the main its properties are proved. Then, by passing to the limit, the integral is extended to the limits of step functions. In which sense are these limits taken? Actually, the Lebesgue integral is defined for the functions that can be expressed as the difference of two functions that are the limits (almost everywhere) of monotonic sequences of step functions. What if we consider only uniform limits (i.e. the functions that are limits of uniformly converging sequences of step functions)? Applying the Cauchy criterion of uniform convergence, we immediately get the convergence of the corresponding sequence of integrals and call its limit the integral of the limit function. The correctness of such a definition is almost obvious. The main properties of the integral (linearity, additivity, the principal formula of integration, etc.) are proved very simply. The class of functions integrable in this sense consists of all functions without discontinuities of the second kind; although this class is narrower than the Riemann-integrable functions, it is sufficient for acquiring the main skills of integration and its applications in other areas (differential equations, mechanics, etc.). Later, moving to the Lebesgue integral is extremely natural by considering – instead of uniformly converging – pointwise (almost everywhere) converging sequences of step functions that are Cauchy sequences in the mean. Matching this scheme with the Daniel scheme is rather simple since the pointwise (a.e.) limit of a Cauchy sequence can be represented as the difference of two limits of monotonic sequences of step functions.
Having indicated the advantages of our approach, it is also worth mentioning one “disadvantage”. When the integral is defined in the way mentioned, the students need to know the relatively difficult notion of uniform convergence of a series of functions, which must be taught perhaps earlier than usual.
Notation
The set of positive integers {1, 2,…}
Real line (–∞, + ∞)
Extended real line
The set of non-negative real numbers [0, + ∞)
∀
“for all”, “for each”, “for every”
∃…:…
There exists … such that…
:=
“Denote”, “is equal by definition to”
≡
Identically equal
“implies”
“if and only if”, “necessary and sufficient”, “equivalent”
Ø
Empty set
x
∈
A
An element
x
belongs to a set
A
x
∉
A
An element
x
does not belong to a set
A
A
c
The complement of a set
A
A
⊂
B
,
B
⊃
A
A set
A
is a subset of a set
B
{
x
n
} ⊂
A
A sequence of elements of a set
A
A
∪
B, A
∩
B
The union and intersection of sets
A
and
B
The union of a sequence of sets
The intersection of a sequence of sets
f: X → Y
A function (mapping) from
X
to
Y
A sequence of functions {
f
n
} uniformly converges to a function
f
(in some given set
A
)
[
x
]
The integer part of a number
x
(the maximal integer not exceeding
x
)
x
∨
y
,
x
∧
y
max{
x, y
}, min{
x, y
}
The indicator of a set for
x ∈ A
c
PART 1
Integration of One-Variable Functions
1
Functions without Second-kind Discontinuities
DEFINITION 1.1.– A function, whereis an interval, is said to be continuous at a point x0 ∈ I if. Otherwise, f is said to be discontinuous (or has a jump) at x0or that x0is a discontinuity point of f.
A discontinuity point x0of a function f is called:
REMARK 1.1.– Because of its generality, the term regular is not perfect for functions without second-order discontinuities. As we will see further, mathematically, a more appropriate term is uniformly measurable; however, it is rather long and inconvenient.
Figure 1.1.A regular function f ∈ D[a, b]
Note that here we consider one-point sets {c} as closed intervals [c, c].
Figure 1.2.A step function φ ∈ S [a, b]
PROPOSITION 1.1.– Let f ∈ C[a, b]. Define the sequence of step functions {φn} by
Then φn in the interval [a, b], that is, the sequence {φn} converges to f uniformly in [a, b].
Figure 1.3.Approximation of a continuous function by step functions
PROOF.– Take arbitrary ε > 0. Since f is uniformly continuous in [a, b] (Cantor theorem), there exists δ > 0 such that
Taking such that , we get that, for all n >N,
REMARK 1.2.– From the proof of proposition 1.1, we can easily see that the uniform convergence φn in the interval [a, b] remains true if (see Figure 1.4):
Figure 1.4.A more general approximation of a continuous function by step functions
THEOREM 1.1.–
REMARK 1.3.– The converse of the first statement is also true: the limit of a uniformly converging sequence of step functions is a function without second-kind discontinuities. Therefore, the class D[a, b] can be characterized as the functions that are uniform limits of sequences of step functions in [a, b].
PROOF.– [of theorem 1.1]
and
Since
by the compactness of the interval [a, b] (finite subcovering property), there exist x1, x2, …, xn ∈ [a, b] such that
or
Suppose, for example, that case (1) is true. Then,
since and .
Thus, for arbitrary ε > 0, we have a function φ ∈ S [a, b] such that
By taking ,…, we get a sequence {φn} ⊂ S [a, b] such that
Clearly, φn in [a, b].
REMARK 1.4 (for a lecturer).– If the latter proof seems to be too difficult for the first-year students, we may omit it (and the whole chapter as well) and later (in Chapter 3) define the integral for functions that are uniform limits of sequences of step functions (that is, in fact, for all functions from D[a, b]), without proving that such functions exhaust the whole class D[a, b]. Of course, in such a case, we need to prove that at least all continuous functions are integrable (proposition 1.1) and, as an easy consequence, so are the functions with finitely many first-kind discontinuities. Such functions are sufficient for developing practical integration skills.
P.1. Problems
PROBLEM 1.1.– Give an example of a function f ∈ D[a, b] with an infinite (countable) set of discontinuity points.
PROBLEM 1.2.– May the set of discontinuity points of a function f ∈ D[a, b] be the rationals of the interval [a, b]?
PROBLEM 1.3.– Does the Dini lemma hold in the class D[a, b], that is, does the monotone convergence D[a, b] ∋ fn ↓ 0 imply that fn 0?
1 If all the functions in a uniformly converging sequence are continuous at some point, so is the limit function.
2
Indefinite Integral
DEFINITION 2.1.– A function1is called a primitive (or anantiderivative) of a function if
PROPOSITION 2.1.–
PROOF.–
DEFINITION 2.2.– The indefinite integral of a functionis the family of its primitives. It is denoted as ∫ f(x)dx. By proposition 2.1,
where F is any primitive of f (provided that it does exist). It is common to write for short
Table of main integrals
PROPOSITION 2.2 (Properties of indefinite integrals).– Suppose that and Then:
provided that both integrals on the right-hand side of the equality exist;
provided that R(φ) ⊂ I and φ is a differentiable function;
provided that f and g are differentiable functions and there exists at least one indefinite integral in the equality.
REMARK 2.2.– Since, strictly speaking, the indefinite integral is not a function, but rather a family of functions, the above equalities need some explanation. For example, the equality in item 1 means the following: if F and G are any primitives of f and g, respectively, then αF + βG is a primitive of the function αf + βg.
PROOF.– 1)
REMARK 2.3.– In practical integration, it is convenient to use the notion of differential. The differential of a differentiable function at a point x ∈ I is the linear function
By the definition of the derivative, we have:
or, for short,
Simplicity and convenience of these formulas is one of the reasons for using, for indefinite integrals, the notation ∫ f(x)dx, rather than ∫ f(x).
EXAMPLES 2.1.–
P.2. Problems
PROBLEM 2.1.– Find the indefinite integrals:
PROBLEM 2.2.– Find the indefinite integrals:
PROBLEM 2.3.– Find the conditions to be satisfied by the coefficients a, b, and c given that the integral:
is a rational function.
PROBLEM 2.4.– Find the indefinite integrals and
PROBLEM 2.5.– Find the indefinite integrals:
PROBLEM 2.6.– Find the indefinite integrals and
PROBLEM 2.7.– Find the indefinite integral .
1 Here and below in this section, is an interval.
3
Definite Integral
3.1. Introduction
The notion of definite integral is closely related to the problem of calculating the area of a geometric figure on the plane. Let us try to calculate the area of the figure that is under the graph of a continuous function over the interval [a, b] (such a figure is called a curvilinear trapezium). 1
Figure 3.1.Calculation of the area of a curvilinear trapezium
Instead of the values of f at the points xk−1, we often take the values at arbitrary points ξk ∈ [xk−1, xk] (Figure 3.2):
[3.1]
Figure 3.2.Modified calculation of the area of a curvilinear trapezium
The narrower the strip, the more exact the value of the area S. We will get the exact value of S passing to the limit as all widths of strips tend to zero:
This limit (not yet defined rigorously) is called the (definite) Riemann integral of the function f in the interval [a, b] (or “from a to b”) and is denoted by . Thus, the definite integral is equal to the area of the corresponding curvilinear trapezium.
The “definition” above may be called “geometric” because it is based on the geometric problem of calculating an area and geometric arguments. We can use another “purely analytic” approach. Let us carefully look at our arguments. First, to define the integral as the area of a curvilinear trapezium, we replaced the curvilinear trapezium by a “close” figure, the union of “narrow” rectangles, the area of which we can calculate. Then, we passed to the limit by taking such figures increasingly close to the trapezium. Analytically, this means that we replaced the function f by a “close” step function φ defined by
Indeed, note that the sum in the approximate equality [3.1] is, in fact, the area of the figure under the graph of φ over the interval [a, b] and, thus, the integral of the function φ in the interval [a, b]: . Taking step functions increasingly “close” to the function f, in the limit, we obtained the integral of the function f.
Thus, the other way of defining the integral is as follows. First, we define the integrals of step functions. Then, we consider the functions that can be approximated by step functions, that is, the functions that can be expressed as the limit of a sequence of step functions. The integral of such a function is then defined as the limit of the corresponding integrals of step functions. In this context, a natural question arises: under which conditions does such a limit exist? Note that the pointwise convergence is not sufficient – for a pointwise converging sequence of step functions, the limit of the corresponding integrals may depend on the choice of the sequence (in similar cases, we speak about the incorrectness of a definition). Therefore, we need some additional conditions on convergence of sequences of step functions. The simplest (and, at the same time, the strongest) condition is the requirement of uniformity of convergence. Since the uniform limits of sequences of step functions are the functions without discontinuities of the second kind (theorem 1.1), we will be able to define the integral namely for such functions. Later, in Chapter 8, replacing the uniform convergence condition by a weaker one, we will be able to extend the class of integrable functions.
DEFINITION 3.1.– Suppose that a functionφ ∈ S [a, b]takes the value yk in the interval . Then the definite integral of the functionφin the interval [a, b] is defined as the sum
[3.2]
where |I| denotes the length of an interval I. It is denoted by.
PROPOSITION 3.1 (The correctness of the definition of the integral).– The value of the sum in equation [3.2] (and thus the integral of a function φ ∈ S [a, b]) does not depend on the choice of a partition of the interval [a, b] by constancy intervals of φ.
PROOF.– Let
The third equality follows from the fact that either ; thus, we always have .
PROPOSITION 3.2 (Elementary properties of the integral of step functions).–
PROOF.– We begin with an important remark. If f and g are two functions from S [a, b
