by Charles Swartz (New Mexico State University)

This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform.

• Measure Theory:
  • Lebesgue Measure
  • Lebesgue-Stieltjes Measure
  • The Nikodym Convergence and Boundedness Theorems
• Integration:
  • Measurable Functions
  • The Lebesgue Integral Convergence in Mean
  • Convergence in Measure
  • Comparison of Modes of Convergence
  • The Radon-Nikodym Theorem
  • The Vitali-Hahn-Saks Theorem
• Differentiation and Integration:
  • Integrating Derivatives
  • Absolutely Continuous Functions
• Introduction to Functional Analysis:
  • Normed Linear Spaces (NLS)
  • Quotient Spaces
  • The Hahn-Banach Theorem
  • Ordered Linear Spaces
• Function Spaces:
  • L^p-Spaces, 1 ≤ p < ∞
  • The Space of Finitely Additive Set Functions
  • The Space of Countably Additive Set Functions
  • The Space of Continuous Functions
  • Hilbert Space
  • and other papers