Foundations of Functional Analysis

  • Friedrich Sauvigny
Part of the Universitext book series (UTX)


We start with the Riemannian integral - and their Riemann integrable functions - and construct a considerably larger class of integrable functions via an extension procedure. Then we obtain Lebesgue’s integral, which is distinguished by general convergence theorems for pointwise convergent sequences of functions. This extension procedure - from the Riemannian integral to Lebesgue’s integral - will be provided by the Daniell integral. The measure theory for Lebesgue measurable sets will appear in this context as the theory of integration for characteristic functions. We shall present classical results from the theory of measure and integration in this chapter, e.g. the theorems of Egorov and Lusin.


Hilbert Space Banach Space Measurable Function Convergence Theorem Weak Convergence 
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Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Friedrich Sauvigny
    • 1
  1. 1.Mathematical Institute, LS AnalysisBrandenburgian Technical UniversityCottbusGermany

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