on August 21, 2000
"Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0 < p < 1.
The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.
Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.
While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!
As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.
on February 24, 2006
One of the problems with modern mathematics is its obsession with rigor which has been attended, over the last few decades, by a mushrooming of symbols and jargon. Much of it is not clearly related to the ideas they serve to label, as evidenced by such terms as the topological use of "filter" whose etymology is obscure (ascribed by some to H. Cartan). Moreover, the particular subject of Lebesgue integration and its generalizations is made even more confusing by a wide variety of approaches depending on an author's penchants--many of whom are enamored with a purely axiomatic approach and who make little or no appeal to intuition or--God forbid!--pictures. The author of the present work is obviously someone who has actually taught mathematics and taught it lovingly. This book is an excellent read with lots of interesting topics well explained from a student's point of view. There seems to be a nice ramping from the truly elementary to the sophisticated, which means the book will interest experienced mathematicians, scientists and engineers. There are lots of "doable" problems that the reader can solve along the way. For the experienced mathematician these little problems help alot as a refresher (Oh!, now I remember, that's how you do it.). I like the emphasis on Euclidean space. Somehow, I always feel more comfortable there! It gives me things I can actually construct and doodle on paper. And, it allows the author to use a few figures in a meaningful way. Which is another of the book's strong points and if I could recommend a future improvement, it would be to bring on more of those pictures! Tristram Needham has done a nice job along these lines with his book "Visual Complex Analysis." (I ordered several copies as Christmas gifts--just kidding!). Anyone who has taught mathematics and genuinely wished to be understood by his students has, at various times, drawn them pictures. Inside the cover sheets are lists of integration formulae, a fourier transform table, and a table of "assorted facts" on things like the Gamma function; which show that this is not only a book on Lebesgue integration but a calculus book with the Lebesgue integral occupying center stage. Everyone who has been enamored by the notion of the integral--as I was as a freshman calculus student and have been ever since--will want to have this book on their shelf.
on September 24, 2003
As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.
The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.
For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library.
on November 24, 2010
This is a very gentle introduction to Lebesgue integration. On the whole it is quite good; however, it does suffer somewhat from a lack of big-picture perspective. So, for example, when defining the measure of compact sets as the infimum of the measures of covering open sets, the one sentence offered by way of motivation is this:
"This procedure is satisfactory because of the topological nature of open sets." (p. 36)
What an utterly useless remark! Of course the definition works because of the nature of open sets, since it is based on open sets. And of course it works because of the topological properties of open sets, since open sets are topological objects and thus have no non-topological properties.
This example is illustrative of the book's general lack of motivation. Readers who want to know the whys behind the definitions will be frustrated throughout. This is completely unnecessary since meaningful motivations for the definition can easily be supplied using material developed later in the book. Doing so would satisfy inquisitive students, bring greater cohesion to the book, illustrate the important and enlightening interplay between examples and definitions, and spare readers from vacuous throwaway remarks about "the topological nature of open sets" and other unintelligent nonsense.
Since Jones choses to ignore this perspective entirely, I shall here supply the motivation missing in his discussion by illuminating his development of measure theory using material from later in the book. Readers who are interested in this type of discussion will be disappointed by Jones' style.
The concept of measure is a generalisation of the concept of length in R, area in R^2, volume in R^3, etc. Thus we must of course define the measure of the empty set to be 0, and the measure of rectangles to be their usual area (I use the language of R^2 for simplicity; it is clear what the analogs are in other dimensions). Just as when we are doing Riemann integration, however, we need only consider "special rectangles," i.e. rectangles whose sides are parallel to the coordinate axes. Special rectangles are convenient since their area is easily expressed in terms of coordinates, and restricting ourselves to special rectangles is not really a restriction anyway (to wit, any area that can be approximated by rectangles can be approximated by special rectangles). Unions of special rectangles are called special polygons and their measure is defined in the obvious way.
Now the measure of any open set can be defined as the supremum of the measures of all special polygons contained within this set. Of course it is intuitively clear that this will give the usual area for any reasonable figure. Furthermore, the restriction to open sets means that nothing very weird can happen, since in any open set there exist, around any point, a ball completely contained in the set. In this ball we can of course fit a special polygon, and thus, in a sense, restricting ourselves to open sets in our definition is a way of making sure that every point of the set is accounted for.
Next we define the measure of a compact set as the infimum of the measures of all the open sets containing it. Why could we not define the measure of a compact set in the same way as for open sets? Such a definition would agree with basic intuition and work well in most examples. It would even work for the Cantor set: this compact set contains no special polygons (i.e., no intervals) so its measure would be zero, as it should be, if we defined the measure of compact sets by approximation from within by special polygons just as we did for open sets. However, we would lose the idea that "ever point is accounted for" if we were to extend this definition beyond open sets. Thus there might be some crazy sets that give unpleasant results with this definition. Indeed, such an example is furnished by a modification of the Cantor set construction that yields so-called "fat" Cantor sets (p. 85). Such a set in fact has positive measure by the true definition, but it too would have measure zero by the definition based on approximation from within by special polygons (since it contains no intervals, for the same reason as the usual Cantor set: any interval is eventually broken up). Thus we see that defining the measure of compact sets by approximation from within by special polygons would lead to unpleasant consequences: for example, the measures of a fat Cantor set and its complement in [0,1] would not add up to 1, which is certainly unpleasant. The definition based on approximation from without by open sets, however, does not lead to such unpleasantries, and, as in the previous stage, we feel that the definition in a sense makes sure that "every point is accounted for," since every point must be covered by the open sets in question.
Moving to general sets, we define the inner measure of any set as the supremum of the measures of all compact sets contained within it, and the outer measure as the infimum of the measures of all the open sets containing it. Except for a small technicality (on which below) we now define the measure of any set to be equal to the inner and outer measure whenever these are equal. If the outer and inner measures are not the same we simply give up and say that the set is not measurable. This only happens for very crazy sets whose existence can only be proved using the axiom of choice. These nonmeasurable sets are the reason why we cannot define the measure of any set in the same way as for compact sets; that is to say, why we cannot just use outer measure as measure and forget about inner measure altogether. For these nonmeasurable sets yield deeply unpleasant results about outer measure, such as the fact that the outer measure of a union of disjoint sets may be smaller than the sum of the outer measures of the parts (p. 83). Thus the purpose of introducing the inner measure is to rule out this sort of predicament.
Nonmeasurable sets are also the basis of the technical point mentioned above. If we take such a set, say in R, and take its union with say (1000, infinity), then we obviously get a set whose inner and outer measures are both infinity. Yet it would be misleading to call this set "measurable," because it can do all sorts of uncontrollable stuff on (-infinity, 1000). To exclude this sort of examples we say that a set of infinite inner measure is called measurable only if its intersection with any measurable set of finite inner measure is itself measurable, and its measure is defined as the supremum of the measures of all such intersections. With the epithet "measurable" defined in this way, the class of measurable sets is conveniently homogenous---e.g. closed under countable unions and countable intersections---in a way that it would not have been if we had admitted those nonmeasurable impostor-sets of infinite inner and outer measure.
on March 29, 2006
This is a terrific text for a first course in graduate-level real analysis, and is suitable for self-study. It develops Lebesgue integration theory slowly, in a very clear manner. In addition, the latter part of the book covers the basics of Fourier Analysis and important topics in differentiation. I frequently refer to this book, as the results are easy to find.
on January 28, 2008
This book is truly marvelous. Unlike many other books written by mathematicians, this one does not focus entirely on the overlycomplicated symbolism that has become so frighteningly common in modern textbooks on pure mathematics. The author, for the most part, demonstrates the concept before moving on to nonconstructive methods. This is accomplished via simple pictures, depicting simple figures such as rectangles. By using these to demonstrate the concepts of set theory, and lebesgue measure, the reader almost certainly will develop some awareness of lebesgue integration. For someone not already familiar with set theory (like myself) I suggest writing down the key terms of set theory (presented in the first 23 pages of the book) and keeping them with you while reading. Mathematical rigorousness should come secondary to understanding the key concepts, and therefore the reader might want to skip over nonessential ideas (usually some kind of proof related to an extension of a key concept) and go right to what is needed to understand what is going on.