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A Course in Analysis, Vol. III: Measure and Integration Theory, Complex-Valued Functions of a Complex Variable

Niels Jacob and Kristian P. Evans
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Jason M. Graham
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A Course in Analysis provides an introduction to real and complex analysis at a level appropriate for advanced undergraduate or beginning graduate students. This is the third volume of the series by Niels Jacob and Kristian P. Evans. As with the first two volumes, the book is based on material for an analysis sequence at Swansea University.

The third volume of A Course in Analysis is subtitled Measure and Integration Theory and Complex-Valued Functions of a Complex Variable. It continues in the spirit of the first two volumes: the authors seek to “to provide a more connected or unified treatment of many of the most important topics in mathematics.” Also as with the first two volumes, this one is written for students and is highly readable.

The book is divided into two parts, with measure theory covered in part I and complex analysis in part II. The treatment of measure theory is carried out via a careful, axiomatic development. Everything is proven in great detail and the approach is useful for students interested in both analysis as well as probability theory. The treatment of complex analysis in part II is equally detailed and also emphasizes the geometric nature of the theory of complex variable functions. In the following we describe in greater detail the contents of the book.

Part I begins with careful definitions of the essential set-theoretic concepts for measures, i.e., \(\sigma\)-field, measurable function, etc. Many examples are provided and some of relevant topological notions are introduced. We note that metric spaces were introduced in volume 2 of A Course in Analysis. The coverage takes the Carathéodory approach. As an application the authors derive the common notion of Lebesgue measure. The authors also introduce Hausdorff measures.

All of the important limit theorems for integrals are given careful treatment as are the Radon-Nikodym theorem and the Lebesgue differentiation theorem. The authors do a very nice job in leading the reader through the technical details of measure theory while simultaneously helping the reader to develop an intuition for the field and some of its typical applications. One could reasonably use part I of the book as a stand-alone course in measure theory and integration.

Part II is where the book really shines. The treatment of complex analysis is beautiful. The authors stress the geometric feel of complex function theory and really help the reader to develop an intuition for complex analysis. Since part II largely does not rely directly on results from part I, the reader could start with part II. Moreover, one could use part II of the book as a text for a course in complex variable theory either before or after an advanced course in real analysis.

Part II begins with a development of the field of complex numbers covering the algebraic, geometric, and analytic facets of \(\mathbb{C}\) in a uniform manner, then proceeds with a detailed and accessible development of complex function theory as far as the proof of the Riemann mapping theorem. As in part I, the authors cover the relevant topology, including a nice treatment of the fundamental group along the way. Even though the mathematics is sophisticated, I believe that students will find the text highly accessible.

Volume 3 of A Course in Analysis is a great book for a first year (U.S.) graduate student. One of the nice features of the book is that the book contains full solutions for all of the problems which make it useful as a reference for self-study or qualifying exam prep. I am looking forward to seeing what the authors have to offer in the upcoming volumes of A Course in Analysis.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

  • Measure and Integration Theory:
    • First Look at σ-Fields and Measures
    • Extending Pre-Measures. Carathéodory's Theorem
    • The Lebesgue-Borel Measure and Hausdorff Measures
    • Measurable Mappings
    • Integration with Respect to a Measure — The Lebesgue Integral
    • The Radon-Nikodym Theorem and the Transformation Theorem
    • Almost Everywhere Statements, Convergence Theorems
    • Applications of the Convergence Theorems and More
    • Integration on Product Spaces and Applications
    • Convolutions of Functions and Measures
    • Differentiation Revisited
    • Selected Topics
  • Complex-Valued Functions of a Complex Variable:
    • The Complex Numbers as a Complete Field
    • A Short Digression: Complex-Valued Mappings
    • Complex Numbers and Geometry
    • Complex-Valued Functions of a Complex Variable
    • Complex Differentiation
    • Some Important Functions
    • Some More Topology
    • Line Integrals of Complex-Valued Functions
    • The Cauchy Integral Theorem and Integral Formula
    • Power Series, Holomorphy and Differential Equations
    • Further Properties of Holomorphic Functions
    • Meromorphic Functions
    • The Residue Theorem
    • The Γ-Function, The ζ-Function and Dirichlet Series
    • Elliptic Integrals and Elliptic Functions
    • The Riemann Mapping Theorem
    • Power Series in Several Variables
  • Appendices:
    • More on Point Set Topology
    • Measure Theory, Topology and Set Theory
    • More on Möbius Transformations
    • Bernoulli Numbers