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The Lebesgue Integral for Undergraduates

The Lebesgue Integral for Undergraduates

William Johnston

Print ISBN: 978-1-93951-207-9
Electronic ISBN: 978-1-61444-620-0
296 pp., Hardbound, 2015
List Price: $60.00
Member Price: $45.00
Series: MAA Textbooks

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The Lebesgue Integral for Undergraduates presents the Lebesgue integral at an accessible undergraduate level with surprisingly minimal prerequisites. Anyone who has mastered single-variable calculus concepts of limits, derivatives, and series can learn the material. The key to this success is the text’s use of a method labeled the “Daniell-Riesz approach.” The treatment is self-contained, and so the associated course, often offered as Real Analysis II, no longer needs Real Analysis I as a prerequisite. Additional curricular options then exist. Academic institutions can now offer a course on the integral (and function spaces) along with Complex Analysis and Real Analysis I, where completion of any one course enhances the other two. Students can enroll immediately after Calculus II, after a first course in mathematical proofs, or as a required course in function theory. Along with Vector Calculus and Probability Theory, this set of courses now provides a comprehensive undergraduate investigation into functions.

The benefits are powerful. The reader now has a gateway into the modern mathematics of functions. At a very early stage, undergraduates now have the required background for collaborative research in function theory. Large numbers of students now have significantly improved access to journal articles in analysis. 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 \(L^2(\mu)\) 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.

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