Week
|
Dates
|
Daily
|
Homework
|
1
|
Aug 27-29
|
Wed:
We'll start with an introduction to the course and
Chapter 1 of WZ.
Fri: We'll continue with Chapter 1 of WZ and
possibly Chapter 3 of WZ. We will skip Chapter 2
for now and come back to it later.
|
Homework #1
Chapter 1: 1klmq, 9, 10, 13, 15, 18.
Due: Fri Sept 5.
Read chapter 1 and get started on chapter 3.
homework
hints
|
2
|
Sept 1-5
|
Mon:
Labor day (no class)
Tues: last day of official add/drop period.
Wed: We studied outer measure.
Fri: We showed the measurable sets form a
sigma-algebra. Homework #1 due.
|
Homework #2
Chapter 3: 4, 5, 9, 12, 13, 17, 20, 24.
Due: Wed Sept 10.
Read chapter 3 and get started on Chapter 4.
homework
hints
|
3
|
Sept
8-12
|
Mon: We showed outer
measure is countably additive when restricted to
measurable sets and proved a continuity theorem
for measures.
Wed: We characterized measurable sets.
Fri: We covered measurable functions.
Lecture
notes: these are outlines of the lectures
sans proofs. Please let me know if you spot an
error. Also your questions are most welcome.
|
Homework #3
Chapter 3: 25
Chapter 4: 3, 5, 6, 8, 12.
Due: Wed Sept 17.
Read Chapter 4 and get started on Chapter 5.
homework
hints
|
4
|
Sept 15-19
|
Mon: We started
Littlewood's 3 principles (measurable sets are
nearly intervals, measurable functions are nearly
continuous and convergent sequences are nearly
uniformly convergent).
Wed: The plan is to cover Lusin's Theorem,
convergence in measure and integrating bounded
functions.
Fri: The plan is to define Lebesgue integration,
prove linearity, and prove the bounded convergence
theorem.
Lecture
notes: these are outlines of the lectures
sans proofs. Please let me know if you spot an
error. Also your questions are most welcome.
|
Homework #4
Chapter 4: 15, 18, 19.
Chapter 5: 2, 4, 9, 13.
Due: Wed Sept 24.
Read Chapter 5.
homework
hints
|
5
|
Sept
22-26
|
Mon: Convergence
Theorems.
Wed: Differentiation.
Fri: We went over the Riemann integral, proved
that the set of compactly supported continuous
functions are dense in L^1, and defined the
maximal function.
|
Homework #5
Chapter 5: 10, 11, 21.
Due: Wed Oct 1.
Read the part of Chapter 2 on functions of bounded
variation and Chapter 7. We'll skip the
Riemann-Stieltjes integral.
homework
hints
|
6
|
Sept 29-Oct 3
|
Mon: We will finish
the proof of Lebesgue's Differentiation Theorem
and start on the proof that monotone functions are
differentiable.
Wed: We will differentiate monotone functions and
functions of bounded variation.
Fri: Exam. The exam will cover chapter 3-5. It
will not have differentiation.
Exam
solutions
|
Homework #6
Chapter 7: 1, 2 (convolution is
defined on page 93), 4.
Chapter 2: 1, 4, 7.
Hint for chapter 7 #2: work out the case when phi
is a scalar multiple of a characteristic function
of a set first.
Due: Wed Oct 8.
Read the part of Chapter 2 on functions of bounded
variation and Chapter 7. We'll skip the
Riemann-Stieltjes integral.
homework
hints
|
7
|
Oct
6-10
|
Mon: Differentiation
of monotone functions.
Wed: Absolutely continuous functions and convex
functions.
Fri: L^p classes.
|
Homework #7
Chapter 7: 6, 10, 11, 12, 14.
Chapter 8: 2.
Due: Wed Oct 15.
Read: Chapters 8 & 10.
homework
hints
|
8
|
Oct
13-17
|
Mon: Banach spaces.
Wed: Hilbert spaces.
Fri: Signed measures and the Hahn Decomposition
(chapter 10).
Lecture
notes: these are outlines of the lectures.
Please let me know if you spot an error. Also your
questions are most welcome.
|
Homework #8
Chapter 8: 4, 5 (just the first question),
6, 11, 12, 13, 14a, 15.
Due: Wed Oct 22.
Read: Chapters 8 & 10. (we'll skip chapter 9
and come back to it later). You might also peruse
the lecture
notes.
homework
hints
|
9
|
Oct
20-24
|
Mon: Jordan
decomposition. (Early
voting begins)
Wed: Radon-Nikodym Theorem (Last day a graduate
student may change registration in a class to or
from the credit/no credit basis.)
12-1 on Wed: homework session in RLM 10.176.
Fri: The dual of L^p. |
Homework #9
Chapter 10: 2, 9, 24.
Due: Wed Oct 29.
Read: 10.1-10.4 and 11.1, 11.2, 11.4. (we're
skipping some optional material). You might also
peruse the lecture
notes.
homework
hints
|
10
|
Oct 27-31
|
Mon: Outer measure
Wed: Caratheodory's Extension Theorem
Fri: Exam 2
Exam
solutions
|
Homework #10: click
here for .pdf.
Due: Wed Nov 5.
Read: 11.1, 11.2, 11.4, chapter 6. (we're skipping
some optional material). You might also peruse the
lecture
notes.
homework
hints
|
11
|
Nov
3-7
|
Mon: Product
measures
Tues: voting
day
Wed: Fubini's Theorem and Tonelli's Theorem
Fri: Fubini's Theorem and Tonelli's Theorem |
Homework #11
Chapter 6: 1, 5, 6, 10.
Chapter 9: 2, 3, 4.
Due: Wed Nov 12.
Read: Chapters 6 and 9 (we're skipping some of 6.3
and we might skip 9.3, 9.4). You might also peruse
the lecture
notes and Arveson's
notes.
homework
hints
|
12
|
Nov 10-14
|
Mon: Convolutions
Wed: Convolutions
Fri: Spaces of measures
|
Homework #12
Chapter 8: 8.
Chapter 9: 5, 6, 7, 10 (hint: see the proof
of 9.16).
Chapter 10: 16, 26.
Due: Wed Nov 19.
Read: Chapter 9 (we might skip 9.3, 9.4). You
might also peruse the lecture
notes and Arveson's
notes.
homework
hints
|
13
|
Nov
17-21
|
|
Homework #13:
click
here for .pdf.
Due: Wed Nov 26.
Read: The last chapter of the book. You might also
peruse the lecture
notes and Arveson's
notes.
homework
hints
|
14
|
Nov 24-28
|
Thurs-Fri:
Thanksgiving
|
Homework #14: click
here for .pdf.
Due: Wed Dec 3.
Read: The last chapter of the book. You might also
peruse the lecture
notes.
homework
hints
|
15
|
Dec
1-5
|
Fri: Last Class day.
Also last day a graduate student or a law student
may, with the required approvals, drop a class or
withdraw from the University.
|
Practice
problems for the final.
(this is not to be turned in).
Some
solutions to the practice problems.
|
16
|
Dec
8-12
|
The final starts
Wednesday Dec 10. I'll email it to you in the
morning. Please return it to my office by Thursday
Dec 11 3pm.
|
The
final (with solutions). |
|
|
Happy Holidays!
|
|