Math 4121, Spring, 2021

Dept. of Mathematics, Washington University

 

Syllabus for Math 4121 

Introduction to the Lebesgue Integral

Instructor: Steven G. Krantz

Classroom: The course will be taught OnLine only. No classroom.

e-mail address: sk@math.wustl.edu

Office: 103, Cupples I

Office Hour: Monday, Wednesday, Friday from 11:00am to 1:00pm, either by email or phone (510-875-8972).

You can send me questions by email at any time.


Phone: (510) 875-8972

Dept. Office: 100, Cupples I

Dept. Phone: (314) 935-6760

Course Home Page: http://www.math.wustl.edu/~sk/math4121.html

 

Course Description: The purpose of this course is

to provide an introduction to the theory of the Lebesgue integral.

We will learn about Lebesgue measure, Lebesgue measurable sets, and the integral.

We will consider many examples to give a solid foundation for the rather abstract ideas in the course.

Applications to other parts of mathematics will be provided.

 

 

Textbook: S. G. Krantz, Elementary Introduction to the Lebesgue Integral,

CRC Press, Boca Raton, FL, 2018.


Elements of the Course:

Homework: 34% of grade

Midterm Exam: (2 of these) each 33% of grade

THERE WILL BE NO FINAL EXAM

TOTAL: 100%

 

The homework is an essential part of this course. Each new idea

builds on previous ideas, so it is essential to master and internalize each

one when you encounter it. Be sure to do the homework in a regular and timely

fashion. Homework should be written out on 8.5" x 11" paper and submitted

on time. Only one problem per page!!! It will be graded. We will use the CrowdMark system for submitting

homework. What you will do then is scan your homework and submit it OnLine.

When a homework assignemnt is formulated, CrowdMark will let you know

and tell you how to submit..

 

Be familiar with this Web page. This is where homework assignments will be

posted, homework solutions posted,

exam solutions posted, due dates will be posted,

and also where exams and other course events and information

will be announced.


We will decide together when our midterm exams will be.

Of course the exams will be at-home and open-book. Usually I give you at least two days to do an exam. If there are particular days when you do not want an exam (for a religious

holiday or other reason), please let me know.


First Math 4121 Lecture, 01-25-2021




Audio for First Math 4121 Lecture, 01-25-2021




Second Math 4121 Lecture, 01-27-2021




Audio for Second Math 4121 Lecture, 01-27-2021




Third Math 4121 Lecture, 01-29-2021




Audio for Third Math 4121 Lecture, 01-29-2021




Fourth Math 4121 Lecture, 02-01-2021




Audio for Fourth Math 4121 Lecture, 02-01-2021




FIRST HOMEWORK ASSIGNMENT. DUE WEDNESDAY, FEBRUARY 3, 11:59PM.


Chapter 1, # 1, 3, 5, 6, 8, 9, 11


SECOND HOMEWORK ASSIGNMENT. DUE FRIDAY, FEBRUARY 12, 11:59PM.


Chapter 2, #1, 2, 4, 5, 6, 8, 9, 11


Fifth Math 4121 Lecture, 02-03-2021




Audio for Fifth Math 4121 Lecture, 02-03-2021




Sixth Math 4121 Lecture, 02-05-2021




Audio for Sixth Math 4121 Lecture, 02-05-2021




Seventh Math 4121 Lecture, 02-08-2021




Audio for Seventh Math 4121 Lecture, 02-08-2021




Eighth Math 4121 Lecture, 02-10-2021




Audio for Eighth Math 4121 Lecture, 02-10-2021




Ninth Math 4121 Lecture, 02-12-2021




Audio for Ninth Math 4121 Lecture, 02-12-2021




Tenth Math 4121 Lecture, 02-15-2021




Audo for Tenth Math 4121 Lecture, 02-15-2021




Eleventh Math 4121 Lecture, 02-17-2021




Audio for Eleventh Math 4121 Lecture, 02-17-2021




Solutions to Homework 1




Twelfth Math 4121 Lecture, 02-19-2021




Audio for Twelfth Math 4121 Lecture, 02-19-2021




Thirteenth Math 4121 Lecture, 02-22-2021




Audio for Thirteenth Math 4121 Lecture, 02-22-2021




Fourteenth Math 4121 Lecture, 02-24-2021




Audio for Fourteenth Math 4121 Lecture, 02-24-2021




THIRD HOMEWORK ASSIGNMENT. DUE MONDAY, FEBRUARY 22, 11:59PM.


Chapter 3, #1, 2, 4, 5, 7, 8, 10, 11, 12


Solutions to Homework 2




Fifteenth Math 4121 Lecture, 02-26-2021




Audio for Fifteenth Math 4121 Lecture, 02-26-2021




Sixteenth Math 4121 Lecture, 03-01-2021




Audio for Sixteenth Math 4121 Lecture, 03-01-2021




Seventeenth Math 4121 Lecture, 03-03-2021




Audio for Seventeenth Math 4121 Lecture, 03-03-2021




FOURTH HOMEWORK ASSIGNMENT. DUE MONDAY, MARCH 8, 11:59PM.


Chapter 4, #1, 2, 3, 5, 7, 8, 9, 10, 12


Solutions to Homework 3




Practice Exam for Midterm 1




Solutions to Practice Exam for Midterm 1




Eighteenth Math 4121 Lecture, 03-05-2021




Audio for Eighteenth Math 4121 Lecture, 03-05-2021




Nineteenth Math 4121 Lecture, 03-08-2021




Audio for Nineteenth Math 4121 Lecture, 03-08-2021




BECAUSE I OVERLOOKED THE WELLNESS DAYS, WE WILL HAVE NO LECTURES ON WEDNESDAY, MARCH 10 AND FRIDAY, MARCH 12. WE WILL START UP AGAIN ON MONDAY, MARCH 15.


FIFTH HOMEWORK ASSIGNMENT. DUE WEDNESDAY, MARCH 24, 11:59PM.


Chapter 5, #1, 3, 6, 9, 13


Chapter 6, #2, 6, 9


Chapter 7, #1, 4 ,6, 8


SIXTH HOMEWORK ASSIGNMENT. DUE MONDAY, APRIL 12, 11:59PM.


Chapter 8, #2, 3, 6, 7


Chapter 9, #1, 3, 6, 10


Chapter 10, #2, 5, 7


Twentieth Math 4121 Lecture, 03-15-2021




Audio for Twentieth Math 4121 Lecture, 03-15-2021




Twenty-First Math 4121 Lecture, 03-17-2021




Audio for Twenty-First Math 4121 Lecture, 03-17-2021




Twenty-Second Math 4121 Lecture, 03-19-2021




Audio for Twenty-Second Math 4121 Lecture, 03-19-2021




First Midterm Exam




Solutions to First Midterm Exam




Twenty-Third Math 4121 Lecture, 03-24-2021




Audio for Twenty-Third Math 4121 Lecture, 03-24-2021




Twenty-Fourth Math 4121 Lecture, 03-26-2021




Audio for Twenty-Fourth Math 4121 Lecture, 03-26-2021




Solutions to Homework 5




Twenty-Fifth Math 4121 Lecture, 03-29-2021




Audio for Twenty-Fifth Math 4121 Lecture, 03-29-2021




Twenty-Sixth Math 4121 Lecture, 03-31-2021




Audio for Twenty-Sixth Math 4121 Lecture, 03-31-2021




Twenty-Seventh Math 4121 Lecture, 04-02-2021




Audio for Twenty-Seventh Math 4121 Lecture, 04-02-2021




Twenty-Eighth Math 4121 Lecture, 04-05-2021




Audio for Twenty-Eighth Math 4121 Lecture, 04-05-2021




Twenty-Ninth Math 4121 Lecture, 04-07-2021




Audio for Twenty-Ninth Math 4121 Lecture, 04-07-2021




Thirtieth Math 4121 Lecture, 04-09-2021




Audio for Thirtieth Math 4121 Lecture, 04-09-2021




Thirty-First Math 4121 Lecture, 04-14-2021




Audio for Thirty-First Math 4121 Lecture, 04-14-2021




SEVENTH HOMEWORK ASSIGNMENT. DUE FRIDAY, APRIL 30, 11:59PM.


Chapter 11, #1, 5


Chapter 12, #1, 4


Chapter 13, #1, 3


Chapter 14, #1, 3


Chapter 15, #3, 7


Chapter 16, #3, 5


A. Let (X, rho) and (Y, sigma) be metric spaces. Describe a method for equipping the set X x Y with a metric manufactured from rho and sigma.


B. Let X be the collection of all continuously differentiable functions on the interval [0,1]. If f,g are elements of X, then define rho(f,g) = sup_{x in [0,1]} |f'(x) - g'(x)| . Is rho a metric? Why or why not?


C. Give an example of a metric space (X, rho), a point P in X, and a positive number r such that {y in X: rho(y, X) <= r} is not the closure of the ball B(P,r).


D. Consider the metric space Q (the rational numbers) equipped with the Euclidean metric. Describe all the open sets in this metric space.


E. Let (X, rho) be the collection of continuous functions on the interval [0,1] equipped with the usual supremum metric. For j a positive integer, let E_j = {p(x): p is a polynomial of degree not exceeding j} . Then, as noted in the lecture, each E_j is nowhere dense in X. Yet the union of the E_j is dense in X. Explain why these assertions do not contradict Baire's theorem.


F. Let p_j(x)$ be a sequence of polynomial functions on the real line, each of degree not exceeding k. Assume that this sequence converges pointwise to a limit function f. Prove that f is a polynomial of degree not exceeding k.


Thirty-Second Math 4121 Lecture, 04-21-2021




Audio for Thirty-Second Math 4121 Lecture, 04-21-2021




Thirty-Third Math 4121 Lecture, 04-23-2021




Audio for Thirty-Third Math 4121 Lecture, 04-23-2021




Thirty-Fourth Math 4121 Lecture, 04-26-2021




Audio for Thirty-Fourth Math 4121 Lecture, 04-26-2021




Practice Exam for Midterm 2




Solutions to Practice Exam for Midterm 2