Graduate Studies in Mathematics, Volume XXX, Amer. Math. Soc., Providence, (to appear).

Topics in Real Analysis

Gerald Teschl

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: 28-01, 42-01
Keywords: Real Analysis, Measure theory, Lebesgue spaces, Fourier transform

The text is available as pdf (1.9M) version. Any comments and bug reports are welcome!
Table of contents
  1. Measures
    1. The problem of measuring sets
    2. Sigma algebras and measures
    3. Extending a premeasure to a measure
    4. Borel measures
    5. Measurable functions
    6. How wild are measurable objects
    7. Appendix: Jordan measurable sets
    8. Appendix: Equivalent definitions for the outer Lebesgue measure
  2. Integration
    1. Integration - Sum me up, Henri
    2. Product measures
    3. Transformation of measures and integrals
    4. Surface measure and the Gauss-Green theorem
    5. Appendix: Transformation of Lebesgue-Stieltjes integrals
    6. Appendix: The connection with the Riemann integral
  3. The Lebesgue spaces Lp
    1. Functions almost everywhere
    2. Jensen ≤ Hölder ≤ Minkowski
    3. Nothing missing in Lp
    4. Approximation by nicer functions
    5. Integral operators
    6. Rearrangements
  4. More measure theory
    1. Decomposition of measures
    2. Derivatives of measures
    3. Complex measures
    4. Appendix: Functions of bounded variation and absolutely continuous functions
  5. Even more measure theory
    1. Hausdorff measure
    2. Infinite product measures
    3. Convergence in measures and a.e. convergence
    4. Weak and vague convergence of measures
    5. The Bochner integral
    6. The Lebesgue-Bochner spaces
  6. The dual of Lp
    1. The dual of Lp, p<∞
    2. The dual of L and the Riesz representation theorem
    3. The Riesz-Markov representation theorem
  7. Sobolev spaces
    1. Warmup: Differentiable and Hölder continuous functions
    2. Basic properties
    3. Extension and trace operators
    4. Embedding theorems
    5. Applications to elliptic equations
  8. The Fourier transform
    1. The Fourier transform on L1 and L2
    2. Some further topics
    3. Applications to linear partial differential equations
    4. Sobolev spaces
    5. Applications to evolution equations
    6. Tempered distributions
  9. Interpolation
    1. Interpolation and the Fourier transform on Lp
    2. The Marcinkiewicz interpolation theorem
    3. Calderón-Zygmund operators
Glossary of notations