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    STUDIA MATHEMATICA - Issue no. 4 / 2010  
         
  Article:   BOOK REVIEWS: JOHN J. BENEDETTO AND WOJCIECH CZAJA, INTEGRATION AND MODERN ANALYSIS, XIX+575 PP,BIRKHÄUSER ADVANCED TEXTS, BIRKHÄUSER, BOSTON -BASEL - BERLIN, 2009, ISBN: 978-0-8176-4306-5, E-ISBN: 978-0-8176-4656-1.

Authors:  ŞTEFAN COBZAŞ. 		 
       
         
  Abstract:  The aim of the present book is to emphasize how the modern integration theory evolved from some classical problems in function theory, related mainly to Fourier analysis. It is worth to mention that some problems that arose in the study of Fourier series in the nineteenth century lay at the basis of many modern branches of mathematics as, for instance, set theory. For this reason the first chapter of the book, Ch. 1, Classical real variables, contains some classical results related to differentiation (e.g., continuous nowhere differentiable functions) and its imperfect relations with the Riemann integral, culminating in the new theory of integration developed by Lebesgue, which put the things in their right places. In fact, one of the main ideas of the book is the study of the relations between integrals and the a.e. derivatives (the Fundamental Theorem of Calculus - FTC), realized by the key notion of absolute continuity, viewed by the authors as a unifying concept for the FTC, Lebesgue dominated convergence theorem (LDC) and Radon-Nikodym theorem.  
         
     
         
         
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