M 547 Fall 2015
Measure Theory
Monday, Wednesday, Friday: 11:00-11:50 am
Wilson 1-147
Prof. Kevin Wildrick
E-mail: kevin.wildrick@montana.edu
Course Website: TBA
Office Location: Wilson 2-223
Office Hours: By appointment
Office Phone: (406) 994-3123
“If people do not believe that mathematics is simple, it is only because they do not realize how
complicated life is.” - John von Neumann
“All things in life are subject to change, including this syllabus.” - Kevin Wildrick
Course Description: This course provides a rigorous introduction to Lebesgue measure on Rn ,
abstract measure theory, and Lebesgue integration. Additional and/or tangential topics may include: Hausdorff measure and the basics of geometric measure theory, Lp -spaces and the basics of
functional analysis, and differentiation theory.
Prerequisite: M 505 Principles of Mathematical Analysis or permission of the instructor.
Credit Hours: 3
Texts: The Elements of Integration and Lebesgue Measure by Robert G. Bartle (primary), and
Real and Complex Analysis by Walter Rudin. These texts are optional and will be used as supplements only; it is my hope that the lectures notes will be self-contained. Copies of the texts will be
available on reserve at the library.
Problem Sets: Weekly problem sets will be due at 11:00 am on Wednesdays.
Exam and Presentation There will be no mid-term exam in this course. Each student has
the option of giving a 25 minute presentation on the topic of their choice (subject to approval) in
exchange for credit on the exam. The amount of this credit will be determined by the quality of
the presentation.
Grading: Exam 60%, Homework 40%.
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Problem set expectations
• Please work in groups. Learning from your fellow students is one of the best things about being
in a graduate program. However, please remember the mantra of mathematical ethics: “It is
unethical to get credit for mathematics that you do not completely understand”.
• Please write up your solutions independently. Writing simple, elegant, and clear mathematics
is an important. It takes a lot of practice to develop this skill, and to develop your own style
of writing. Even if you solved the problem with a group, write up your own solution alone.
• Be very critical of your proofs, and be honest with yourself about their veracity. I would
much rather read “I cannot prove (insert statement here). If it is true, then ... ” than
“(insert statement here) is true because (insert unclear nonsense here).”
• Feel free to use outside references to help you solve problems. However, your solutions are
expected to be essentially self-contained. This means that, for the most part, only theorems
and results that have already been presented in the lectures or M505 can be used as blackboxes. Of course, this is not a hard and fast rule - you will have to use your judgement when
using results from other courses or your previous studies. A good rule of thumb is “If you
can’t explain the proof of a theorem on demand, don’t use it as a black-box in your solutions.”
• Only you can decide how much detail to provide. One way of doing this is to think of a
solution as a letter to a slightly younger student
• Always write in grammatically correct prose. Treat displayed formulas and symbols as grammatical entities. Write complete sentences.
– Acceptable: Since the function f is L-Lipschitz continuous, we see that
L1 (f (E)) ≤ LL1 (E)
for any measurable subset E ⊆ R, as shown in Theorem 4.2 in class.
– Not acceptable:
f L-Lipschitz → L1 (f (E)) ≤ LL1 (E)
(Theorem 4.2).
Failure to write grammatically correct prose will result in a confused and grumpy grader who
is less inclined to be generous with mathematical mistakes.
• Solutions must be typed, preferably in LaTeX. Please ask me or a friend for a template and
a tutorial if you are unfamiliar with LaTeX.
• Late solutions will not be accepted. However, one problem set will be dropped from the
calculation of grades.
Most importantly, have fun! Frustration and being “stuck” are part of the game -learn
to enjoy them and you will never suffer while doing math.
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