Overview

Using the Daniell–Riesz approach, this text presents the Lebesgue integral at a level accessible to an audience familiar only with limits, derivatives and series. Employing such minimal prerequisites allows for greatly increased curricular flexibility for course instructors, as well as providing undergraduates with a gateway to the powerful modern mathematics of functions at a very early stage. The book's topics include: the definition and properties of the Lebesgue integral; Banach and Hilbert spaces; integration with respect to Borel measures, along with their associated L2(μ) spaces; bounded linear operators; and the spectral theorem. The text also describes several applications of the theory, such as Fourier series, quantum mechanics, and probability.

Full Product Details

9781939512079

Table of Contents

Reviews

In 1902, modern function theory began when Henri Lebesgue described a new 'integral calculus.' His 'Lebesgue integral' handles more functions than the traditional integral--so many more that mathematicians can study collections (spaces) of functions. For example, it defines a distance between any two functions in a space. This book describes these ideas in an elementary, accessible way. Anyone who has mastered calculus concepts of limits, derivatives, and series can enjoy the material. Unlike any other text, this book brings analysis research topics within reach of readers even just beginning to think about functions from a theoretical point of view. - Mathematical Reviews Clippings When I noticed the title of this book, I was curious to see if this subject actually could be made comprehensible to an undergraduate. It turns out that it really can be, via a path to the Lebesgue integral that is different from the one I took as a graduate student. The approach used here is one of Daniell and Riesz, and avoids the development of a lot of technical measure theory. The author asserts in the Preface that the text is accessible to anyone who has mastered the single-variable calculus concepts of limits, derivatives and series. Of course, claims of broad accessibility such as this are frequently made and often exaggerated, but here the author is, if you add integration to the list of calculus-related concepts that are necessary, most of this book really is pretty accessible to the audience described. The author achieves this level of accessibility by employing a number of useful pedagogical features in the text. First and foremost is a very reader-friendly style that I found to be slow, careful, conversational, well-motivated and clear. A careful selection of homework exercises also enhances the pedagogical value of the text; in fact, the author provides various different kinds of exercises. Some, for example, are integrated right into the text, with the intention of having the student answer them as encountered. Solutions to these appear at the end of each chapter. I like books that try something new, offer a different perspective on things, and are carefully and clearly written. This one qualifies on all counts. This is a book, I think, that students will actually read, and even better, enjoy. - Mark Hunacek, MAA Reviews

"In 1902, modern function theory began when Henri Lebesgue described a new 'integral calculus.' His Lebesgue integral handles more functions than the traditional integral--so many more that mathematicians can study collections (spaces) of functions. For example, it defines a distance between any two functions in a space. This book describes these ideas in an elementary, accessible way. Anyone who has mastered calculus concepts of limits, derivatives, and series can enjoy the material. Unlike any other text, this book brings analysis research topics within reach of readers even just beginning to think about functions from a theoretical point of view."" - Mathematical Reviews Clippings ""When I noticed the title of this book, I was curious to see if this subject actually could be made comprehensible to an undergraduate. It turns out that it really can be, via a path to the Lebesgue integral that is different from the one I took as a graduate student… I like books that try something new, offer a different perspective on things, and are carefully and clearly written. This one qualifies on all counts. This is a book, I think, that students will actually read, and even better, enjoy."" - Mark Hunacek, MAA Reviews"

Author Information

William Johnston is Professor of Mathematics at Butler University, Indiana. His publications include articles on operator theory and functional analysis, and the undergraduate textbooks A Transition to Advanced Mathematics: A Survey Course (with Alex McAllister) and An Introduction to Statistical Inference.

Tab Content 6

Customer Reviews

Recent Reviews

Countries Available