MATH 5322, Section 001, 1:00 p.m. MWF, Room MA115


REAL ANALYSIS, Modern Techniques and Their Applications, by G.B. Folland, 2nd Ed.


Name: Alex Wang
Office: MA237
Office Hours: 10:00-11:00 am, M-F.
Office Phone: 834-7626

Expected Learning Outcomes

Upon completion of this two-semester series, students should master concepts and theories of outer measure, the Caratheodory extension theorem, general measures, Lebesgue integrals with respect to a measure, Lebesgue measures, Lebesgue-Stieltjes measures, product measures, convergence theorems, Fubine-Tonelli theorem, signed measures, functions of bounded variation, absolutely continuous functions, differentiation theory, differentiation of a measure, metric spaces, compactness, Banach spaces, Lp spaces, Hilbert spaces, basic Fourier analysis, bounded linear functionals, dual spaces, and bounded linear operators.

Grading Policy


Homework problems are assigned weekly at The homework score is determined by

sum of the scores of all assignments

total available points
× 500

Midterm Tests:

Three midterm tests of 100 points each will be administered on the following days:
Test 1: Friday, September 22.
Test 2: Friday, October 20.
Test 3: Friday, November 17.

Final Exam:

The final exam is comprehensive and will be worth 200 points. It will be administered on Wednesday, December 13, from 1:30 p.m. to 4:00 p.m.

Grading Scale:

The total points will be calculated as the sum of the scores of the Homework, Midterm Tests, Final Exam, and Optional Projects. The course grade will be determined by
A: 900 or above
B: 800-899
C: 700-799
D: 600-699
F: 0-599

Critical Dates

September 22: Test 1
October 20: Test 2
October 30: Last day to drop
November 17: Test 3
December 13: Final Exam 1:30 p.m.

Tips on Mathematical Proof:

  1. Use connection phrases before and between your statements to identify it, whether is a hypothesis, an assumption, a derivation, or a conclusion. Never write a statement without identify it.

    The connection phrases for hypothesis are: "given", "by the given condition", "we know", "if", "since", "because", "assuming", "by definition" etc.

    The connection phrases for derivation are: "then", "so", "it implies that", "therefore", "hence", "it follows that", "accordingly", "thus", "consequently", etc.

    Examples of other connection phrases are: "also", "besides", "furthermore", "in addition to", "likewise", "moreover", "similarly", "conversely", "in other words", "in particular", "otherwise", "although", "alternatively", "however", "nevertheless", "on the other hand", "actually", "certainly", "clearly", "obviously", "in fact", "in deed", "surely", etc.

  2. Define your notation either before using it ("Let A=......") or immediately after using it ("where A=......").

  3. Use quantifiers, never write an open sentence. There are only two quantifiers: ∀ and ∃. Do not use "let" at the place of a quantifier. "Let" is used to assign a name or notation. "Let x∈S, P(x)" is incorrect and should never be used.

  4. In Mathematics, every statement is wrong until proving correct. Never write any statement you can not justify.

  5. Never write an irrelevant statement. Never write a definition if you do not use it immediately.

  6. Never assume what you want to prove to be true before proving it.

  7. Never prove a statement with the quantifier "∀" by showing that it is true for a few examples.

  8. The best way to prove an existence type result is to find one, or give a procedure to construct one.

Absence due to religious observance: The Texas Tech University OP 34.19 states that a student who intends to observe a religious holy day should make that intention known in writing to the instructor prior to the absence. A student who is absent from classes for the observance of a religious holy day shall be allowed to take an examination or complete an assignment scheduled for that day within a reasonable time after the absence. As your instructor, I request that notification be made in writing and submitted no later than the 15th class day of the semester. Absence due to officially approved trips - The Texas Tech University OP 34.04 states department chairpersons, directors, or others responsible for a student representing the university on officially approved trips must notify the student's instructors of the departure and return schedules. The instructor so notified must not penalize the student, although the student is responsible for material missed. Any student absent because of university business must be allowed to make up missed work within a reasonable span of time or have alternate grades substituted for work due to an excused absence. Students absent because of university business must be given the same privileges as other students.

Academic Integrity (extracted from OP 34.12): It is the aim of the faculty of Texas Tech University to foster a spirit of complete honesty and high standard of integrity. The attempt of students to present as their own any work not honestly performed is regarded by the faculty and administration as a most serious offense and renders the offenders liable to serious consequences, possibly suspension. Scholastic dishonesty includes, but it not limited to, cheating, plagiarism, collusion, falsifying academic records, misrepresenting facts, and any act designed to give unfair academic advantage to the student (such as, but not limited to, submission of essentially the same written assignment for two courses without the prior permission of the instructor) or the attempt to commit such an act.

Civility in the Classroom: Incivility is any action that interferes with the classroom learning environment. This includes, but is not limited to arriving late, leaving early, not closing/putting aside the cell phone, text messaging, sleeping, chatting during class, disturbing others with noise, dominating the class discussion. Be respectful to the instructor and to your fellow students, or you may be asked to leave the classroom.

Accommodation for Students with Disabilities (extracted from OP 34.22): Any student who, because of a disability, may require some special arrangements in order to meet course requirements should contact the instructor as soon as possible to make the necessary arrangements. Students should present appropriate verification from Student Disability Services during the instructor's office hours. Please note instructors are not allowed to provide classroom accommodations to a student until the appropriate verification from Student Disability Services has been provided. For additional information, you may contact the Student Disability Services office at 335 West Hall or 806-742-2405.