21-720: Measure and Integration     Fall 2021

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:

Recommended reference works:

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:

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.