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Birkhäuser - Birkhäuser Mathematics - Scuola Normale Superiore | Introduction to Measure Theory and Integration

Introduction to Measure Theory and Integration

Ambrosio, Luigi, Da Prato, Giuseppe, Mennucci, Andrea C.G.

1st Edition., 198p.

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  • Good choice of exercises, supplemented with solutions
  • Modern view point on measure theory and integration
  • Classical results and proofs
This textbook collects the notes for an introductory course in measure theory and integration. The course was taught by the authors to undergraduate students of the Scuola Normale Superiore, in the years 2000-2011. The goal of the course was to present, in a quick but rigorous way, the modern point of view on measure theory and integration, putting Lebesgue's Euclidean space theory into a more general context and presenting the basic applications to Fourier series, calculus and real analysis. The text can also pave the way to more advanced courses in probability, stochastic processes or geometric measure theory. Prerequisites for the book are a basic knowledge of calculus in one and several variables, metric spaces and linear algebra. All results presented here, as well as their proofs, are classical. The authors claim some originality only in the presentation and in the choice of the exercises. Detailed solutions to the exercises are provided in the final part of the book.

Content Level » Graduate

Keywords » Fourier series - calculus - measure theory - real analysis - space theory

Related subjects » Scuola Normale Superiore

Table of contents 

Measure spaces.- Integration.- Spaces of measurable functions.- Hilbert spaces.- Fourier series.- Operations on measures.- The fundamental theorem of integral calculus.- Operations on measures.- Appendix: Riesz representation theorem of the dual of C(K) and integrals depending on a parameter.- Solutions to the exercises.

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