This is a course in the theory of measures and integrals defined by measures.
Roughly, a measure is
a way of associating a non-negative number to certain subsets (called measurable
sets) of a set X in such a way that the whole is equal to the sum of the
parts in the sense that the measure of any countable disjoint union of
measureable sets is the sum of the measures. Measure theory provides
the theoretical basis for all of
the tools used in probability
and statistics. Lebesgue measure on R^n reduces on rectangles in
R^n to the usual n-dimensional volume = product of the side lengths.
For functions from R^n
to R which are Riemann integrable, the value of the Lebesgue integral coincides
with that for the Riemann integral. But there are many more
Lebesgue integrable
functions than Riemann integrable functions and there's an explicit sense
in which Lebesgue integration completes Riemann integration. Modern
modeling applications commonly use measure and integration theory on abstract
spaces with essentially no restrictions on measurement functions instead
of old-fashioned R^n models and restriction to Riemann integrable functions.
Time and Place: Tuesday, Thursday, 10:00-11:30 a.m., in Room 200, Cupples II
Instructor: Edward
N. Wilson
Office: Cupples I, Room 18 (in the basement)
Tentative Office Hours: TTh 1:30-3:00 and by appointment
Office Tel: 935-6729 (has voice-mail)
E-mail: enwilson@math.wustl.edu
Prerequisites: Math 4111 is a hard and fast prerequisite. Either Math 309 or Math 429 is desirable along with a "nodding" acquaintance with probablity and statistics.
Textbook: The Elements of Integration and Lebesgue Measure, Robert Bartle, Wiley paperback edition, 1995.
Topic Outline: We'll
go through Bartle's book fairly sytematically. The hope will be to
get through essentially the entire book (174 pages) by the end of the semester.
Chapters 1-8 and 10 are devoted to abstract measure spaces and associated
integrals while Chapters 9 and 11-17 restrict attention to Lebesgue measure
on R^n. Possibly we'll do Chapter 9 (construction of Lebesgue measure)before
some of Chapters 6-8 in order to be able to better illustrate the "power"
of the general theory. If time permits, we'll talk about a few additional
topics: Fourier series and Fourier integrals, convolution tools,
change of variable theorem for Lebesgue integrals,...
Exams/Homework: There
will be a take-home exam in late March and an in-class final exam during
finals week. There will usually be a homework assignment to be handed
in each week
with most of the problems
coming from Bartle's problem sections.
Grading: Final
averages will be determined by the following formula:
Final average = .30MidtermExam + .40FinExam + .30HW
Thus, the mid-semester
exam will be 30% of the final average, the final exam 40%, and homework
30%. When homework scores are good and one exam score is significantly
higher than the other, a different formula may be used to lessen the influence
of the lower
score. Also, the process
of converting final averages to letter grades won't, from a student's point
of view, be any worse than the traditional 90-100 A, 80-90 B, 70-80 C scale
but might well be better, i.e., more generous.
Academic Integrity:
As
with all Washington University courses, cheating on exams will be
taken very seriously
with evidence supporting a cheating allegation forwarded to the Arts
and Sciences Integrity
Committee for adjudication. When the Committee concludes that a
student cheated on an
exam, it normally directs the instructor to assign the student a
failing grade for the
course.
Cheating on homework consists of either blindly copying off someone else's
solutions
or not acknowledging
the receipt of assistance from others in completing the assignment.
It's not anticipated
that students will work in isolation on homework problems. To the
contrary, discussing
problems with others is often a way to avoid frustration and gain
useful insight. However,
all students are expected to write up their own assignments and
to indicate in a short
note at the top of the first page the names of any people (other
than the instructor)
with whom they discussed the problems or from whom they received some
hints. Violation
of these requests will result in an instructor-imposed penalty (e.g.,
something like half
credit for the assignment) but won't be treated as a "hanging"
offense--in particular,
won't be brought to the attention of the Arts and Sciences
Integrity Committee.