Main Text:

• R. G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley Classics, 1995.

• Paul R. Halmos, Measure Theory, Springer-Verlag, 1974. (Classic.)
• Walter Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1987. (A classic but very terse treatment, including a variety of other topics.)
• G. B. Folland, Real analysis. Modern Techniques and Their Applications 2nd ed. Wiley-Interscience, 1999. (A useful collection of extra topics.)
• R. L. Wheeden and A. Zygmund, Measure and Integral, CRC Press, 1977.
• N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Wiley Interscience. (Huge. Chapter III deals with integration, including Bochner integrals.)

Course Description:   This course treats deals with the Lebesgue integral in $\R^n$ in particular and the abstract theory of integration and measures in general. Topics covered include:

• Measurable sets, sigma-algebras. Measurable functions. Measures, measure spaces.
• Integrals and convergence theorems: monotone convergence, Fatou's lemma, Dominated convergence.
• Lebesgue measure, Borel measures, outer measure, Carath\'eodory's extension theorem
• Modes of convergence, theorems of Egoroff and Lusin
• Hahn and Jordan decompositions, Radon-Nikodym derivative. Lebesgue's differentiation theorem.
• Product measures, Fubini's theorem.
• $L^p$ spaces, Riesz representation theorem. Holder and Minkowski inequalities, completeness, equiintegrability, Vitali convergence theorem.

Prerequisites: Real analysis (21355), linear algebra (21341), metric-space topology (21465). (Course numbers are only examples.)

Grading:   Based on

• Homework (about 6 sets)
• Mid-term exam
• Final exam

Homework PDF files will be posted online at http://www.math.cmu.edu/~bobpego/21720/ .

Final Exam: There will be a mandatory comprehensive final examination; time and date to be announced.

Make-up tests are only possible in the case of a documented medical excuse, a university-sanctioned absence (e.g., participation in a varsity sporting event), or a family emergency. Please contact me at the earliest time possible to schedule a make-up.