MTH 241, Fall 1999
Real Analysis
MWF 9:10 - 10:00 218 Physics
Greg Lawler, 128B Physics
Office Hours: Mondays and Thursdays, 1:30 - 3:00, and by
appointment
Text
Royden, Real Analysis. This will be supplemented by other
sources for probability and Fourier analysis. I have
put two books on reserve in the math-physics
library: Adams and Guillemin, Measure Theory
and Probability and Billingsley, Probability and Measure.
Syllabus
This will be a course in measure theory. We develop simultaneously the
theory of Lebesgue measure/integration on the reals and the measure theory
needed to do probability. Standard results about convergence and inequalities
involving integrals will be done. There will also be an introduction to
Fourier analysis including a proof of the central limit theorem using Fourier
analysis. Here
is a tentative syllabus for the class.
Assignments
There will be weekly homework assignments. They will be due on
Wednesdays. An assignment on a particular Wednesday will be on
material covered the previous week, so that it can be worked
on over the weekend and questions asked on Mondays. Late
homeworks will not be accepted. Homework assignments will be
available here; if problems outside the text are assigned
the problem sets will be in PDF format.
Problem Set 1 (due September 8)
Problem Set 2 (due September 15) p. 64, 10, 13; p. 258, 3;
p. 291, 2; p. 298, 3,7.
Problem Set 3 (due September 22) p. 50, 48; pp. 70-72, 19, 21,
24, 25, 28; p.263, 14.
Problem Set 4 (due September 29)
Problem Set 5 (due October 6)
Problem Set 6 (due October 13)
Problem Set 7 (due October 27) p.101, 1, 2; p. 104, 7, 10;
pp. 110-112, 12, 20; pp. 116-117, 23, 27
Problem Set 8 (due November 3) p. 119, 2; p.123, 8;
pp. 126-127, 10, 12, 17; pp. 218-219, 5
Problem Set 9 (due November 10)
Problem Set 10 (due November 17)
Problem Set 11 (due December 8)
Additional Notes
On occasion, I will supply additional notes for material
that is not in Royden. They will be available here
in pdf format.
Probability Spaces (posted Septmber 10)
Random Variables and Expectation (posted September 20)
Independence (posted October 4)
Sums of Independent Random Variables (posted October 4)
Central Limit Theorem (posted November 9)
Conditional Expectation (posted November 26)
Note added August, 2000: I have combined the notes above and
corrected some typos.
Here
are all the notes
in a single pdf file.
For material on Hilbert space and Fourier series, the
recommended reading is 3.2 - 3.5 of Adams and Guillemin
Exam
There will be a closed book, in-class final examination. There
will be no other exams during the semester. Grades will be based
on homeworks and the final exam.
Return to: Greg Lawler's home
page
Last modified: 25 August 2000