MAT 523 -- Real Variables I 
Fall 2000 Course Description

Text: Gerald B. Folland, "Real Analysis: Modern Techniques and their Applications," (2nd ed.) John Wiley and Sons, New York, 1999. Chapters 0 through 4 and possibly 5.

References:

• Halsey L. Royden, "Real Analysis," (3rd ed.) MacMillan, New York, 1988. Chapters 1-12.
• Paul R. Halmos, "Measure Theory," Springer Graduate Texts in Mathematics, New York, 1950.  Chapters I-VII, and the "Prerequisites".
• Walter Rudin, "Real and Complex Analysis," McGraw-Hill, New York, 1987.  Chapters 1-8.

Syllabus:

• Prologue
• Set Theory, Cardinality, Orderings
• Axiom of Choice and its Equivalents
• Metric Spaces
 
• Measure Theory
• Sigma-Algebras, Measure Spaces
• Premeasures and Outer Measures
• Construction of Borel Measures on R and Lebesgue-Stieljes Measures
• Properties of Lebesgue Measure
 
• Integration
• Measurable Functions
• Integration of Nonnegative, Real- and Complex-Valued Functions
• Convergence Theorems, Comparison with the Riemann Integral
• Modes of Convergence
• Product Measures - Fubini & Tonelli Theorems
• The Lebesgue Integral on Rⁿ
 
• The Daniell Integral
• Topology
• Topological Spaces, Continuous Maps
• Urysohn's Lemma, the Tietze Extension Theorem
• Nets and Filters
• Compactness; Tychonoff's Theorem
• The Stone-Weierstrass Theorem
 
• Introduction to Functional Analysis
• Normed Vector Spaces
• Linear Functionals; the Hahn-Banach Theorem
• Baire Category
• Hilbert Spaces