Professor: Robert Pego     Email: rpego AT cmu.edu   Office phone: (412)268-2553   Office: 6127 Wean Hall   Office hours: MWF 4:00-5:00pm (usually) and by arrangement

Lectures:   MWF 2:30-3:20 pm in Wean Hall 4625. My rough lecture notes will be made available via the course Canvas page. If it becomes necessary, recordings of class sessions may be made available via the Canvas page, for students' personal, educational use only.

Main Texts:

• R. G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley Classics, 1995.
    (Chapters available online via CMU libraries. We start with chapter 11.)
• G. B. Folland, Real analysis. Modern Techniques and Their Applications 2nd ed. Wiley-Interscience, 1999.  
    (link to latest errata)

• S. Axler, Measure, Integration & Real Analysis, Springer, 2020. (new work by a fine writer, open access.)
• E. M. Stein and R. Shakarchi, Real Analysis, Princeton, 2005. (A thorough, basic exposition by a master analyst.)
• T. Tao, An Introduction to Measure Theory, AMS, 2011. (Many unique insights.)
• 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.)
• N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Wiley Interscience. (Huge. Chapter III deals with integration, including Bochner integrals.)

Course overview:   This course treats the Lebesgue integral in Rⁿ in particular and the abstract theory of integration and measures in general. These theories are fundamental in many areas of mathematics and probability. E.g., Gian-Carlo Rota has suggested that measure theory and probability could be called ``continuous combinatorics.'' Topics to be addressed 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, Carathéodory'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. Hölder and Minkowski inequalities, completeness, equiintegrability, Vitali convergence theorem.
• Radon measures, duality.

Learning objectives: Students should gain an understanding of the fundamental concepts of measure theory and master proofs of the fundamental theorems underlying the theory of integration. They should become familiar with the nature of Lebesgue function spaces, duality, and various important notions of convergence. Also they will gain the ability to use the concepts, methods, and results of measure theory to address a variety of problems in analysis.

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

Grading: Based on approx. 8 homework sets (30%), two midterm exams (20% each) and a final exam (30%). Late homework will not receive a grade. But if there is a valid reason for not completing a particular homework set (e.g., illness), it will not count towards your grade. In addition, the lowest homework score will be dropped.

Homework: In addition to reviewing lecture notes and reading book material, actively working exercises is essential for mastering this subject. PDF files will be posted on the course Canvas page. Solutions should be scanned and uploaded into Gradescope via link from Canvas.

Academic integrity requires that your tests and homework solutions are your independent work and not copied from other sources. On homework you are encouraged, however, to discuss with others and consult other resources to improve your understanding. I find that explaining to someone else is the best way to consolidate my own understanding!

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.

Health and wellness: Graduate study is time-consuming and can be stressful. Make your health a priority, be smart about time management, seek social support, and ask for help when needed.