A Course in Analysis

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

In this third volume of "A Course in Analysis", two topics indispensible for every mathematician are treated: Measure and Integration Theory; and Complex Function Theory.

In the first part measurable spaces and measure spaces are introduced and Caratheodory's extension theorem is proved. This is followed by the construction of the integral with respect to a measure, in particular with respect to the Lebesgue measure in the Euclidean space. The Radon–Nikodym theorem and the transformation theorem are discussed and much care is taken to handle convergence theorems with applications, as well as Lp-spaces.

Integration on product spaces and Fubini's theorem is a further topic as is the discussion of the relation between the Lebesgue integral and the Riemann integral. In addition to these standard topics we deal with the Hausdorff measure, convolutions of functions and measures including the Friedrichs mollifier, absolutely continuous functions and functions of bounded variation. The fundamental theorem of calculus is revisited, and we also look at Sard's theorem or the Riesz–Kolmogorov theorem on pre-compact sets in Lp-spaces.

The text can serve as a companion to lectures, but it can also be used for self-studying. This volume includes more than 275 problems solved completely in detail which should help the student further.

Contents:

• 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

Readership: Undergraduate students in mathematics.
Key Features:

• A coherent treatment of the entire core in mathematical analysis for undergraduate and graduate students — a unique source for learning as well as a reference text for later studies
• A comprehensive and modern treatment of two indispensible theories for every mathematician: measure and integration theory, complex function theory
• More than 275 problems solved in full detail