Notice that for the counting measure on X = N the σ -algebra is X = P (N). So, if each section E n belongs to Y, we have that the set { n } × E n. The Elements of Integration and Lebesgue Measure has 27 ratings and 2 reviews. afloatingpoint said: 5/28/ So far: A very rigorous text! Robert G. Bartle. Bartle Elements of Integration and Lebesgue Measure – Ebook download as PDF File .pdf) or read book online.

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Full text of ” Bartle, R. Wiley Classics Library Edition Published Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section or of the United States Copyright Act without the permission of the copyright owner is unlawful. Library of Congress Cataloging in Publication Data: It is possible to read these two parts in either order, with only a bit of repetition.

The Elements of Integration is essentially a corrected reprint of a book with that title, originally published indesigned to present the chief results of the Lebesgue theory of integration to a reader hav- ing only a modest mathematical background. This book developed from my lectures at the University of Illinois, Urbana-Champaign, and it was subsequently used there and elsewhere with considerable success.

We suppose that the reader has some familarity with the Riemann integral so that it is not necessary to provide motivation and detailed discussion, but we do not assume that the reader has a mastery of the subtleties of that theory.

Sherbert provides an adequate background. In preparing this new edition, I have seized the opportunity to correct certain errors, but I have re- sisted the temptation to insert additional material, since I believe v vi Preface that one of the features of this book that is most appreciated is its brevity.

The Elements of Lebesgue Measure is descended from class notes written to acquaint the reader with the theory of Lebesgue measure in the space R p. The main ideas of Lebesgue measure are presented in detail in Chaptersalthough some relatively easy remarks are left to the reader as exercises. The final two chapters venture into the topic of nonmeasurable sets and round out the subject.

There are many expositions of the Lebesgue integral from various points of view, but I believe that the abstract measure space approach used here strikes directly towards the most important results: Further, this approach is particularly well- suited for students of probability and statistics, as well as students of analysis. Since the book is intended as an introduction, I do not follow all of the avenues that are encountered.

However, I take pains not to attain brevity by leaving out important details, or assigning them to the reader. Readers who complete this book are certainly not through, but if this book helps to speed them lebesguf their way, it has accomplished its purpose. In the References, I give some books that I believe readers can profitably explore, as well as works cited in the body of the text.

I am indebted to a number meaeure colleagues, past and present, for their comments and suggestions; I particularly wish to mention N.

The Elements of Integration and Lebesgue Measure

I also wish to thank Professor Roy O. Davies of Leicester University for pointing out a number of errors and possible improve- ments. Introduction Reasons for the development of the Lebesgue integral, comparison with meaxure Riemann integral, the extended real number system 2.

Measurable Functions Measurable sets and functions, combinations, complex- valued functions, functions between measurable spaces 3. Measures Measures, measure spaces, almost everywhere, charges 4.

Integrable Functions Integrable real-valued functions, positivity and linearity of the integral, the Lebesgue Dominated Convergence Theorem, integrands that depend on a parameter 6. Volumes of Cells and Intervals Cells, intervals, length, cells in R pp-dimensional volume, translation invariance Examples of Measurable Sets Borel sets, null sets, translation invariance, existence of non-Borel sets Approximation of Measurable Sets Approximation by open sets, approximation by closed sets, approximation by compact sets, approximation by cells Additivity and Nonadditivity Additivity, Caratheodory revisited, inner measure x Contents The later work of Newton and Leibniz enabled this method to grow into a systematic tool for such calculations.


As this theory developed, it has become less concerned with applica- tions to geometry and elementary mechanics, for which it is entirely measurw, and more concerned with purely analytic questions, for which the classical theory of integration is not always sufficient. Thus a present-day mathematician is apt to be interested in the con- vergence of orthogonal expansions, or in applications to differential equations or probability.

For him the classical theory of integration which culminated in the Riemann integral has been largely replaced by the theory which has grown from the pioneering work of Henri Lebesgue at the beginning of this century.

The reason for this is very simple: Although this enlargement is useful in itself, its main virtue is that the theorems relating to the interchange of the limit and the integral are valid under less stringent assumptions than are required for the Riemann integral. Since one 2 The Elements of Integration frequently needs to make such interchanges, the Lebesgue integral is more convenient to deal with than the Riemann integral.

Once again, this inference follows easily from the Lebesgue Dominated Convergence Theorem. At the risk of oversimplification, we shall try to indicate the crucial difference between the Riemann and the Lebesgue definitions of the integral.

Recall meaeure an interval in the set R of real numbers is a set which has one of the following four forms: In each of these cases we refer to a and b as the endpoints and prescribe Introduction 3 b — a as the length of the interval. The Lebesgue integral can be obtained by a similar process, except that the collection of step functions is replaced by a larger class of functions.

In somewhat more detail, the notion of length is generalized to a suitable collection X of subsets of Lwbesgue.

Once this is done, the step functions are replaced by simple functions, which are finite linear combinations of characteristic functions of sets belonging to X. The integral can then be extended to certain functions that take both signs. Although the generalization of the notion of length to certain sets in R which are not necessarily intervals has great interest, it was observed in by Maurice Frechet that the convergence properties of the Lebesgue integral are valid in considerable generality.

Indeed, let X be any set in which there is meassure collection X of subsets containing the empty set 0 and X and closed under complementation and countable unions. In this case an integral can be defined for a lebesbue class of real-valued functions on X, and this integral possesses strong convergence properties. As we have stressed, we are particularly interested in these con- vergence theorems.

Therefore we wish to advance directly toward them in this abstract setting, since it is more general and, we believe, conceptually simpler than the special cases of integration on the line or in R n. However, it does require that the reader temporarily accept the fact that interesting special cases are subsumed by the general theory.

Specifically, it requires that he accept the assertion that there exists a countably additive measure function that extends the notion of the length of an interval. The proof of this assertion is in Chapter 9 and can be read after completing Chapter 3 by those for whom the suspense is too great. In this introductory chapter we have attempted to provide motivation and legesgue set the stage for the detailed discussion which follows.


Some of our remarks here have been a bit vague and none of them has been proved. These defects measire be remedied. However, since we shall have occasion to refer to the system of extended real meaxure, we now append a brief description of this system. It is stressed that these symbols are not real numbers.

If the limit inferior and the limit superior are equal, then their value is called the limit of the sequence. It is clear that this agrees with the conventional definition when the sequence and the limit belong to R. In various applica- tions the set X may be the unit interval 7 — [0, 1] consisting of all real numbers.

Since the development of the integral does not depend on the character of the underlying space X, we shall make no assumptions about its specific nature. To be precise, we shall assume that this family contains the empty set 0 and the entire set X, and that X is closed under complementation and countable unions.

An ordered pair X, X consisting of a set X and a a-algebra X of subsets of X is called a measurable space. Any set in X is called an 6 Measurable Functions 1 A-measurable set, but when the a-algebra X is fixed as is generally the casethe set will usually be said to be measurable.

The reader will recall the rules of De Morgan: We shall now give some examples of a-algebras of subsets. It is readily checked that X 3 is a a-algebra. We observe that there is a smallest a-algebra of subsets of X containing A. To see this, observe that the family of all subsets of X is a a-algebra containing A and the intersection of all the a-algebras containing A is also a meazure containing A. This smallest a-algebra is sometimes called the a-algebra generated by A.

The Borel algebra is the a-algebra B generated by all open intervals a, b in R. Observe that the Borel algebra B is also the a-algebra generated by all closed intervals [a, b] barle R. Any set in B is called a Borel set. It is readily seen that B is a a-algebra and it will be called the extended Borel algebra. The next lemma shows that measude could have modified the form of the sets in defining measurability.

The following statements are equivalent for a function f on X to R: Since B a and A a are complements of each other, statement a is equivalent to statement b. Similarly, statements c and d are equivalent. Hence a implies c. Therefore, it belongs to B. Certain simple algebraic combinations of measurable functions are measurable, as we shall now show. Let f and g be measurable real-valued functions and let c be a real number.

It barlte clear that 2. The preceding discussion pertained to real-valued functions defined on a measurable space.

Full text of “Bartle, R. G. The Elements Of Integration And Lebesgue Measure”

However, in dealing with sequences of measurable functions we often wish to form suprema, limits, etc. Hence we wish to define measurability for extended real-valued functions and we do this exactly as in Definition 2. Measurable Functions 1 1 The following lemma is often useful in treating extended real-valued functions. Hence f is measurable. It is a consequence of Lemmas 2. We shall return to the measurability of the product fg after the next result.

Let g m be meaasure similarly. It follows from Lemma 2. Let n be a fixed natural number. It is readily established that the properties abc hold.

It is easy to see that sums, products, and limits of complex-valued measurable functions are also measurable. Although this definition of measurability appears to differ from Definition 2.