Modern Analysis provides coverage of real and abstract analysis, offering a sensible introduction to functional analysis as well as a thorough discussion of measure theory, Lebesgue integration, and related topics. This significant study clearly and distinctively presents the teaching and research literature of graduate analysis:

  • Providing a fundamental, modern approach to measure theory
  • Investigating advanced material on the Bochner integral, geometric theory, and major theorems in Fourier Analysis Rn, including the theory of singular integrals and Milhin's theorem - material that does not appear in textbooks
  • Offering exceptionally concise and cardinal versions of all the main theorems about characteristic functions
  • Containing an original examination of sufficient statistics, based on the general theory of Radon measures
    With an ambitious scope, this resource unifies various topics into one volume succinctly and completely. The contents span basic measure theory in an abstract and concrete form, material on classic linear functional analysis, probability, and some major results used in the theory of partial differential equations. Two different proofs of the central limit theorem are examined as well as a straightforward approach to conditional probability and expectation.
    Modern Analysis provides ample and well-constructed exercises and examples. Introductory topology is included to help the reader understand such items as the Riesz theorem, detailing its proofs and statements. This work will help readers apply measure theory to probability theory, guiding them to understand the theorems rather than merely follow directions.
  • chapter 1|15 pages

    Set Theory and General Topology

    chapter 3|16 pages

    Banach Spaces

    chapter 4|14 pages

    Hilbert Spaces

    chapter 5|23 pages

    Calculus in Banach Space

    chapter 8|14 pages

    The Abstract Lebesgue Integral

    chapter 9|22 pages

    The Construction Of Measures

    chapter 10|18 pages

    Lebesgue Measure

    chapter 11|14 pages

    Product Measure

    chapter 12|16 pages

    The Lp Spaces

    chapter 13|26 pages

    Representation Theorems

    chapter 14|22 pages

    Fundamental Theorem of Calculus

    chapter 15|20 pages

    General Radon Measures

    chapter 16|19 pages

    Fourier Transforms

    chapter 17|52 pages


    chapter 18|16 pages

    Weak Derivatives

    chapter 19|16 pages

    Hausdorff Measures

    chapter 20|32 pages

    The Area Formula

    chapter 21|16 pages

    The Coarea Formula

    chapter 22|40 pages

    Fourier Analysis in ℝn

    chapter 24|26 pages

    Convex Functions