MATH 4121 Introduction to Lebesgue Integration
Spring, 2008
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 “size description” 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.
More precisely, we insist that the difference of two measureable
sets is measurable and that a countable disjoint union of measureable
sets is measureable with the measure of the union
being the sum of the measures of the constituent parts. We then use an
abstract version of the “area under the graph” idea to define
integrals with respect to a measure.
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,
Instructor: Edward N. Wilson
Office: Cupples I, Room 18 (in the basement)
Tentative Office Hours: TTh
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 probability 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 in the order listed below. 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 much of Chapter 9 and all of Chapters 11-17 restrict
attention to Lebesgue measure on R^n.
In fact, the book is essentially the merger of two sets of lecture notes, one (Chapters
1-10) doing the “big theorems” on abstract measure spaces and the other
(Chapters 11-17) doing only the R^n case and
repeating much of what was done in Chapter 9.
I.
Overview. We’ll expand on the fairly murky
overview given in Chapter 1 and try to make clear how probability theory is a
special case of general measure theory.
II.
Definitions
of Measures and Measurability plus a few easy warm-up lemmas (Chapters 2 and
3).
III.
Construction
of Measures (Chapter 9 plus relevant bits and pieces of Chapters 11-17). The big theorem is the Caratheodory
extension theorem. We’ll use it to
explicitly describe not only Lebesgue measure on the Borel sets in R^n but describe
via distribution functions every measure on the Borel
sets which is finite on compact sets.
IV.
Definition
of Integrals and Proofs of the Basic Convergence and Completeness Theorems
(Chapters 4-7). This material is the “heart”
of measure and integration theory.
V. Radon-Nikodym Theorem and its use in establishing the Change of Variable Theorem for Lebesgue integrals on R^n (Chapter 8 plus a number of related things not in the text). We won’t dot all the i’s and cross the t’s in the proof of the general Change of Variable Theorem since the proof is long and very technical.
VI.
General
Fubini Theorem (Chapter 10).
VII.
Assorted
Results on Lebesgue Integrals on R^n
(mostly not in the textbook). We’ll
do as much of this as time permits. Possibilities include the properties of
convolutions, the Stone-Weierstrass Theorem, Fourier
series and Fourier integrals, and some modern applications of this material.
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
and Sciences Integrity
Committee for adjudication. When the Committee concludes that a student cheated on an exam, it normallydirects 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.