Book
Graduate Studies in Mathematics, Volume XXX,
Amer. Math. Soc., Providence, (to appear).
Topics in Real Analysis
Gerald Teschl
Abstract
This manuscript provides a brief introduction to Real Analysis.
It covers basic measure theory including Lebesgue and Sobolev spaces and the Fourier transform.
There is also an accompanying text on Functional Analysis.
MSC: 2801, 4201
Keywords: Real Analysis, Measure theory, Lebesgue spaces, Fourier transform
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Table of contents

Preface
 Measures
 The problem of measuring sets
 Sigma algebras and measures
 Extending a premeasure to a measure
 Borel measures
 Measurable functions
 How wild are measurable objects
 Appendix: Jordan measurable sets
 Appendix: Equivalent definitions for the outer Lebesgue measure
 Integration
 Integration  Sum me up, Henri
 Product measures
 Transformation of measures and integrals
 Surface measure and the GaussGreen theorem
 Appendix: Transformation of LebesgueStieltjes integrals
 Appendix: The connection with the Riemann integral
 The Lebesgue spaces L^{p}
 Functions almost everywhere
 Jensen ≤ Hölder ≤ Minkowski
 Nothing missing in L^{p}
 Approximation by nicer functions
 Integral operators
 Rearrangements
 More measure theory
 Decomposition of measures
 Derivatives of measures
 Complex measures
 Appendix: Functions of bounded variation and absolutely continuous functions
 Even more measure theory
 Hausdorff measure
 Infinite product measures
 Convergence in measures and a.e. convergence
 Weak and vague convergence of measures
 The Bochner integral
 The LebesgueBochner spaces
 The dual of L^{p}
 The dual of L^{p}, p<∞
 The dual of L^{∞} and the Riesz representation theorem
 The RieszMarkov representation theorem
 Sobolev spaces
 Warmup: Differentiable and Hölder continuous functions
 Basic properties
 Extension and trace operators
 Embedding theorems
 Applications to elliptic equations
 The Fourier transform
 The Fourier transform on L^{1} and L^{2}
 Some further topics
 Applications to linear partial differential equations
 Sobolev spaces
 Applications to evolution equations
 Tempered distributions
 Interpolation
 Interpolation and the Fourier transform on L^{p}
 The Marcinkiewicz interpolation theorem
Glossary of notations
Index