| Date |
Material covered |
HW problems and
assignments |
Remarks |
| Aug 24 |
Preliminaries on sets
The Cantor-Schröder-Bernstein Theorem
|
|
|
| Aug 26 |
Countable sets
Cantor's diagonal argument
Well ordered sets
|
Homework 1 (tex) |
Homework is due Friday, September 2, by 6:00pm |
| Aug 29 |
Comparability of well ordered sets
The axiom of choice
|
|
|
| Aug 31 |
Preliminaries on Metric spaces
Continuity and completeness
|
|
|
| Sep 2 |
Compactness
The Bolzano-Weierstrass property
The Heine-Borel property
|
Homework 2 (tex) |
Homework is due Friday, September 9, by 6:00pm |
| Sep 5 |
Normed spaces
Banach spaces
Bounded continuous functions
|
|
|
| Sep 7 |
Measurable spaces
Vitali sets
|
|
|
| Sep 9 |
Measurable functions
Pointwise limits
Simple functions
|
|
|
| Sep 12 |
Basic properties of measure spaces
Outer measures
|
Homework 3 (tex) |
Homework is due Monday, September 19, by 6:00pm |
| Sep 14 |
Carathéodory's extension theorem
|
|
|
| Sep 16 |
Lebesgue-Stieltjes measures
|
|
|
| Sep 19 |
The Cantor set
The Cantor function
|
|
|
| Sep 21 |
Regularity of Borel measures
|
Homework 4 (tex) |
Homework is due Wednesday, September 28, by 6:00pm |
| Sep 23 |
Lusin's theorem
|
|
|
| Sep 26 |
Definition of the integral
Integrable functions
|
|
|
| Sep 28 |
Properties of the integral
|
|
|
| Sep 30 |
The monotone convergence theorem
Fatou's lemma
|
|
|
| Oct 3 |
The dominated convergence theorem
|
|
|
| Oct 5 |
Product measures
|
|
|
| Oct 7 |
Fubini's theorem
|
Homework 5 (tex) |
Homework is due Monday, October 17, by 6:00pm |
| Oct 10 |
Lebesgue measure on Euclidean spaces
|
|
|
| Oct 12 |
Midterm Exam
|
Exam
|
Solutions
|
| Oct 17 |
Signed measures
Hahn decomposition
Jordan decomposition
|
|
|
| Oct 19 |
Complex measures
Total variation
|
|
|
| Oct 21 |
Lebesgue's decomposition theorem
The Radon-Nikodym theorem
|
|
|
| Oct 24 |
Polar decomposition for a complex measure
|
|
|
| Oct 26 |
Dual of L1
|
|
|
| Oct 28 |
Topological spaces
Separation axioms
|
|
|
| Oct 31 |
Continuity
Nets
|
|
|
| Nov 2 |
Urysohn's lemma
The Tietze extension theorem
|
Homework 6 (tex) |
Homework is due Wednesday, November 9, by 6:00pm |
| Nov 4 |
Compact spaces
Tychonoff's theorem
|
|
|
| Nov 7 |
The Banach-Alaoglu theorem
The Arzelà-Ascoli theorem
|
|
|
| Nov 9 |
The Stone-Weierstrass theorem
|
Homework 7 (tex) |
Homework is due Wednesday, November 16, by 6:00pm |
| Nov 11 |
Stone-Čech compactification
|
|
|
| Nov 14 |
Urysohn's metrization theorem
|
|
|
| Nov 16 |
The Baire category theorem
|
Homework 8 (tex) |
Homework is due Wednesday, November 30, by 6:00pm |
| Nov 18 |
Cantor space
|
|
|
| Nov 28 |
The Cantor-Bendixson theorem
|
|
|
| Nov 30 |
Polish spaces
|
Homework 9 (tex) |
Homework is due Wednesday, December 7, by 6:00pm
No late homework
|
| Dec 2 |
Suslin scheme's
Lusin's Separation Theorem
|
|
|
| Dec 5 |
Kuratowski's theorem on standard Borel spaces
|
|
|
| Dec 7 |
Standard probability spaces
|
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|
| Saturday, Dec 17: 3:00pm |
Final Exam
|
Exam
|
Solutions |