Benedetto J.J., Czaja W. Integration and Modern Analysis

Basel: Birkhäuser, 2009. — 593 p.A paean to twentieth century analysis, this modern text has several important themes and key features which set it apart from others on the subject. A major thread throughout is the unifying influence of the concept of absolute continuity on differentiation and integration. This leads to fundamental results such as the Dieudonné–Grothendieck theorem and other intricate developments dealing with weak convergence of measures.Key FeaturesFascinating historical commentary interwoven into the exposition;
Hundreds of problems from routine to challenging;
Broad mathematical perspectives and material, e.g., in harmonic analysis and probability theory, for independent study projects;
Two significant appendices on functional analysis and Fourier analysis.
Key Topics:
In-depth development of measure theory and Lebesgue integration;
Comprehensive treatment of connection between differentiation and integration, as well as complete proofs of state-of-the-art results;
Classical real variables and introduction to the role of Cantor sets, later placed in the modern setting of self-similarity and fractals;
Evolution of the Riesz representation theorem to Radon measures and distribution theory;
Deep results in modern differentiation theory;
Systematic development of weak sequential convergence inspired by theorems of Vitali, Nikodym, and Hahn–Saks;
Thorough treatment of rearrangements and maximal functions;
The relation between surface measure and Hausforff measure;
Complete presentation of Besicovich coverings and differentiation of measures.
Integration and Modern Analysis will serve advanced undergraduates and graduate students, as well as professional mathematicians. It may be used in the classroom or self-study.Classical Real Variables
Lebesgue Measure and General Measure Theory
The Lebesgue Integral
The Relationship between Differentiation and Integration on $$ \mathbb{R}$$
Spaces of Measures and the Radon–Nikodym Theorem
Weak Convergence of Measures
Riesz Representation Theorem
Lebesgue Differentiation Theorem on $$ \mathbb{R}^d $$
Self-Similar Sets and Fractals
Functional Analysis
Fourier Analysis