General Information for Math 7311, Real Analysis, Fall 2011
Time MWF at 11:40 AM -- 12:30 PM
Location Room 113 Lockett
Leonard Richardson Office 386 Lockett
Office Hours M -- F, 12:40--1:30 PM. I am available at many other times. Call first to make sure I'm able meet with you. I answer email many times daily.
Telephone 578-1568
Graduate Assistant Ms. Yunyun Yang will grade your homework and hold office hours on Mondays from 12:30 -- 1:30 in 327 Lockett Hall.
Optional Problem Session Fridays starting at 2:30 PM in Room 284 Lockett. It is not necessary to stay the whole time if you do not have questions. This is to go over homework problems and other questions in measure and integration in an informal setting. It is also a good opportunity to help with variations in preparation since students have not all had the same preparation in undergraduate analysis/advanced calculus. I have reserved the room until 4:30 in case there are many questions, but probably we will finish in only an hour or so.


Measure and Integration: A Concise Introduction to Real Analysis, Leonard F. Richardson, 2009. ISBN: 978-0-470-25954-2. You will find a current list of Errata and Addenda at that site.


Advanced Calculus (undergraduate analysis), such as Math 4031 and 4032, and/or Math 4035, or by permission of the Department. Mathematics graduate students will have been placed into Math 7311 by the Graduate Director after reading the student's transcript for evidence of readiness. Graduate students from external departments need to be very sure to take the Advanced Calculus prerequisites, since such courses are rarely included in undergraduate programs in engineering or the applied sciences. If there is any doubt whatever, be sure to consult either with me or with the Graduate Director in Mathematics.

Assignments, Tests, and Grades

Several problems will be collected approximately once per week. These assignments will collectively be the main basis for your final grade. However, since most students taking this Core Course are also preparing to take the department's Comprehensive Exam (which serves as both the PhD Qualifying Exam and the MS Final Exam for a PhD-Preparatory MS), it is desirable that you have some experience with in-class examinations. Thus we will also have a Mid-term Exam and a Final Exam. These tests may include any question (or variation thereupon) in a covered section of the text, whether the exercise was collected for homework or not. As with Qualifying Exams, no books or notes may be used during the test. However, greater weight will be given to the homework assignments in determining Final Grades for this course. I do not give a formula for calculating Final Grades, based on average scores for example, because I believe such formulas are inadequate at this level. (Comprehensive Exams also do not have a fixed passing score - this varies from exam to exam.) An A in this course designates work at a level indicative of successful doctoral study, whereas a B designates work at a level indicative of successful study at least in the Masters Program. A and B are the only satisfactory grades for graduate students. Of course these judgments are subjective, but I believe they provide students with better insight than a formula-based grade. Students are welcome to discuss with me privately how their work looks at any time during the semester. In fact, I encourage every graduate student to do so.

General Advice: 7 Keys to Success!

  • Do not miss class! There are many reasons why students miss classes: Almost all of these are extremely foolish or reflect outright negligence. This class resembles an old-fashioned one-room school house. There are students of many different backgrounds enrolled. Some have prior graduate experience, even including an MS degree in mathematics from another institution, whereas many others have come directly from a BS program. Students are not in competition with one another for the better grades. Ask questions. It is both your right and your responsibility to learn and to understand.
  • It is proper for this course to discuss homework problems with one another, to offer and receive hints or suggestions, and, of course, to ask me for hints or suggestions. Indeed, it is much better to do this than to throw up ones hands in despair and decide one cannot do the homework! If I give a significant hint to a student, I will also email the hint to the whole class for fairness.
  • You must acknowledge help received, or ideas gleaned from books, just as a researcher must do in writing a paper. It is from the efforts we make that we learn, and it is learning at a sufficiently deep level that really matters here.
  • Memorizing the statements of theorems is a shallow form of learning. What matters is understanding the ideas behind the proofs---that is, what it is that makes each theorem true. And it is very important to build a repertoire of examples that illustrate why the hypotheses of each theorem are necessary by showing how the theorem can fail when the hypotheses are not satisfied. This is how you should read and how you should think about homework problems.
  • When you find you are missing something, which is a very common occurrence and should not be a cause for embarrassment, you should ask questions! Asking questions is not a weakness. On the contrary, it is an important virtue. Good research begins with asking questions!
  • Do not expect to be able to do the homework in only two hours per class day! Graduate study should be regarded as being at least 50% more than a full time job.
  • Do not leave the homework for the day before the problems are due! Start doing them as soon as they are assigned. You should expect to return repeatedly to your solutions to find errors and make improvements before handing them in. Hand in your homework on time! Remember that the grader is a fellow graduate student, and his or her time is valuable too.

Homework Assignments

We will maintain a running list of assignments and tests as the course unfolds. The assignments listed in the table below are from 2008. When I have updated an assignment for fall 2011 I will place a due date in the left-hand column.
Academic Honesty The University has clear policies requiring academic honesty. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question. If on the other hand you get a good idea from another book, or from talking with a friend, academic honesty requires that you acknowledge your sources openly. This is a sound principle which will serve you well throughout your life. Moreover, on a practical level, it is very foolish claim as your own an argument from a former student in this class or from a textbook. The arguments which are copied can be recognized very easily as not coming from the student, and often the precise source can be identified readily. This means that the honorable course of action is also the practical one.
Due Date Assignments: The student should do most of the problems in this book. Any of the problems in the text might appear on an examination. This is especially important for those seeking to pass PhD Qualifying Exams. But hand in only the problems in red for grading. The assignments listed currently are from 2008. They will be updated for 2011 when a due date appears in the first column.
Aug 22 Today is the first day of classes. You should have your texts for your courses and be ready for work!
Aug 24 Read the introductory pages in the text and also Chapter 1. Make sure you can do problems 1.1-1.5. These won't be collected but it is necessary to understand them. Turn in a Time Chart today if you would like to participate in an optional 4th hour each week for questions about homework problems or advanced calculus background.
Aug 26Make sure you can do 2.1-2.10, but hand in 2.4 and 2.7.
Sept 2Hand in 2.15, 2.17, and 2.18.
Sept 9Hand in 2.19, 2.20, and 2.21.
Sept 19Hand in 3.3, 3.6, 3.8. Also, be sure you can solve 3.1, 3.2, 3.4, 3.5, 3.7, 3.9.
Sept 26Hand in 3.11, 3.13. Also, be sure you can solve 3.12, 3.14.
Oct. 3Hand in 3.18, 3.19, 3.21, and 4.2. Not to hand in: 3.16, 3.17, 3.23 -- 3.27, 4.3, 4.4.
Oct 10Hand in 4.5, and 4.6, 4.8. Also, 4.7. Note that problem 4.6 is rather subtle. Remember that arbitrary unions of Lebesgue null sets need not be Lebesgue null sets!
Oct 12Midterm Exam. For a guide as to format, here is a sample test: the one used as the Midterm Exam, Fall 2008.
Oct 15Midterm Exam. Here is the midterm exam, with the overall class statistics on page 2: Midterm Exam, Fall 2011.
Oct 17 Hand in 4.10, 4.12. Also, 4.9, 4.11, 4.13. Problem 4.14 is an optional BONUS Problem worth 30 points added to your total homework score for the term. If you decide to do this problem, hand it in on October 21.
Oct 24Hand in 5.5, 5.6, 5.7. Be able to solve 5.8, 5.9, 5.11--5.13.
Oct 31Be able to solve 5.16-5.19; 5.21 5.23, 5.24, 5.26--30. Hand in 5.20, 5.22--5.23, 5.25.
Nov 7Be able to solve 5.33--5.35, 5.39--5.40, 5.43. Hand in 5.32, 5.41, 5.42 5.45.
Nov 14Be able to solve 6.2-6.3, 6.9, 7.6. Hand in 6.4, 6.5, 6.6 (Hints strongly recommended), 6.7(a--d only). These problems are all applications of Fubini's theorem.
Nov 21Be able to solve 6.8, 7.8, 7.9. Hand in 6.10, 6.11, 6.20. Also hand in the new exercise that is Item 20 on the errrata/addenda page This item number changed Saturday because of the insertion of two typo corrections a few hours before. It is the exercise about two ways that knowing the integral over each interval determine f, for insertion on P. 131, since that is where we needed to use it. For problem 6.11, you should not use the result of problem 3.26. The reason I did not assign 3.26 to be handed in is because you would be proving that theorem in problem 6.11!
Nov 23 The following is a 20 point optional Bonus Problem: You may hand in to me on Wednesday before Thanksgiving for 20 bonus homework points the solutions to 6.9a together with 5.43c. But this new two part bonus problem (6.9a plus 5.43c) is not a required assignment---it is optional. You may also make it a 30-point optional bonus if you do the new second part for 6.20 that is described in item #19 on the Errata sheet. Also, I plan to hold this week's optional question session on Wednesday afternoon at 2:30 instead of Friday this week, since Friday is part of the Thanksgiving holiday, but probably most of us will still be here on Wednesday afternoon.
Nov 28Be able to solve 7.10, 7.12, 7.14--7.15. Hand in 7.11, 7.13, 7.16.
Nov 26 -- Dec 5 Please download a copy of the 2011 Take-Home Final Examination. Be sure to read and follow the instructions carefully! Ask me if you have any questions.
Dec 5 The Take-Home Final Exam is available to download and print above. Read and follow the instructions carefully as printed at the top of the test. The test may be handed in to me any time Monday morning, December 5, until noon. If you leave the test in my letter box Monday morning, in case you don't find me in my office, please place it into a sealed envelope first. The test will be an open book test, but you must do it yourself. It is different from homework, because on the take home exam you are required to work without consulting one another. You may however ask questions to me and I may then send hints to the whole class as appropriate via email.