Math 6100 - Theory of Functions of a Real Variable - Fall 2016

Lecture Notes (Updated November 8, 2016)
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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

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
Saturday, Dec 17: 3:00pm Final Exam Exam Solutions

Mathematical resources:

Math department events.
Preliminary exams materials (password protected).
Zentralblatt Math.
Essential Mathematical LaTeX.
Mathematics Stack Exchange.


Career advice from Terence Tao.
Advice to a young mathematician from Sir Michael Atiyah, Béla Bollobás, Alain Connes, Dusa McDuff, and Peter Sarnak.
Ten lessons I wish I had been taught, by Gian-Carlo Rota.
Words of encouragement from Mark Sapir.

Interesting links:

David Hilbert radio address.
An interview with Andrei Kolmogorov.
John von Neumann, a documentary.
John von Neumann at the dedication of the NORD.
My analysis prelim from UCLA.
Oberwolfach photo collection.
Math problems.
Banach-Tarski, from the students at the University of Copenhagen.
Tom Lehrer.
Find an Erdös number.
History of mathematics.
The Mathematics Genealogy Project.
Video of Georg Cantor.