MAT 523 -- Real Variables I 
Fall 1998 Course Description

Text: Halsey L. Royden, "Real Analysis," (3rd ed.) MacMillan, New York, 1988. Chapters 1-6, 11 & 12.

References:

• Gerald B. Folland, "Real Analysis: Modern Techniques and their Applications," John Wiley and Sons, New York, 1984.  Chapters 1-3, 5 & 6, and the "Prologue".
• Paul R. Halmos, "Measure Theory," Springer Graduate Texts in Mathematics, New York, 1950.  Chapters I-V, VI & VII, and the "Prerequisites".
• Walter Rudin, "Real and Complex Analysis," McGraw-Hill, New York, 1987.  Chapters 1-3, 7 & 8.

Syllabus:

• Measure Theory
• Sigma-Algebras
• Measureable Functions
• Measure Spaces
• Construction of Lebesgue Measure and its Properties
• Product Measures - Fubini & Tonelli Theorems
• Signed Measures - Hahn Decomposition and Radon-Nikodym Theorems
• Integration
• Abstract Integration
• Convergence Theorems
• The Lebesgue Integral
• Comparison with the Riemann Integral
• Differentiation
• Derivatives
• Absolutely Continuous Functions
• Functions of Bounded Variation
• Fundamental Theorem of Calculus for the Lebesgue Integral
• Functional Analysis
• L^p Spaces
• Hölder and Minkowski Inequalities
• Riesz-Fischer Theorem: Completeness of L^p
• Bounded Linear Functionals
• Isometries of L^p
• Riesz Representation Theorem