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).

**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 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

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

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

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

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

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

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.