Material

• We will pretty much cover chapters 0-5 and parts of chapter 6 of: Gerald B. Folland, Real analysis: modern techniques and their applications, 2nd ed. In particular:
  
  
  1. Measures
  - Sigma-algebras
  - Measures, outer measures
  - Borel measures on the real line, non-measurable sets
  
  2. Integration
  - Measurable Functions
  - Littlewood's three principles
  - Integration of Nonnegative Functions
  - Integration of Complex Functions
  - Modes of Convergence
  - Product Measures
  - The n-dimensional Lebesgue Integral
  - Integration in Polar Coordinates
  
  3. Signed Measures and Differentiation
  - The Hardy-Littlewood maximal function
  - Signed Measures
  - The Lebesgue-Radon-Nikodym Theorem
  - Complex Measures
  - Differentiation on Euclidean Space
  - Functions of Bounded Variation
  
  6. L^P spaces
  
  
• Useful reading, for Real Analysis I and Real Analysis II:
  • Folland, G.B. (1984). Real Analysis, New York, Wiley.
  • Roydan, H.L. (1969). Real Analysis, New York, MacMillan
  • Rudin, W. (1986). Real and Complex Analysis, New York, McGraw Hill (3rd edition).
  • Stein, E.M. and Sharkarchi, R. (2005). Real Analysis, Measure Theory, Integration and Hilbert Spaces, Princeton University Press.
  • Wheeden, R. and Zygmund, A. (1977). Measure and Integral, An Introduction to Real Analysis, New York, M. Dekker.
  • Rudin, W. (1976). Principles of Mathematical Analysis, New York. McGraw Hill. (3rd edition).
  • Stein, E.M. and Sharkarchi, R. (2003). Fourier Analysis, Princeton University Press.