This page gives an approximate schedule of the topics that will be covered in lecture and a listing of related problems. Section numbers and problems numbers refer to the text by Folland unless indicated otherwise.
M Jan 8 Lecture 1 Introduction/Overview
Review of metric spaces and basic topology (0.6,4.1)
Assignment: Read over chapter 0
due Jan 17 Read carefully 0.1 Sets (ask questions on Wednesday)
1 Problems
Read 1.1-2 Sigma-Algebras
W 10 Lecture 2 Construction of a nonmeasurable set
1.2-5 The Basic Lebesgue/Caratheodory Construction
Assignment: Read 0.2 Orderings
Jan 17 Read 1.3 Measures
Problems 1.2.1,2
M 15 Holiday
w 17 Lecture 3 Some details from section 1.2
(come back to product sigma-algebras later)
More details from section 1.3
Assignment: Read 0.3 Cardinality
Jan 22 Read 1.4 Outer Measure
2 Problems 1.2.3a,4,5
M 22 Lecture 4 1.3 Measures, Outer Measure, and Sets of Measure Zero
Assignment: Read 0.4 Transfinite Induction
Jan 29 Problem
3 Problems 1.3.8,9,10 (come back to local measurability and
saturation later; 1.3.16,21,22)
W 24 Lecture 5 Review 1.3 Properties of Measures
1.4 The Caratheodory measurability condition (part I)
Assignment: Read 0.5 Extended Real Numbers
Jan 29 Problem
M 29 Lecture 6 Review 1.3 Properties of Measures
1.4 The Caratheodory measurability condition (part II)
Motivational Examples/Inner Measure
Assignment: Read 0.6 Metric Spaces
Feb 5 Problem
4 Problem 1.4.17
W 31 Lecture 7 1.4 The Caratheodory measurability condition (part III)
Properties of Measurable Sets
The Restriction/Extension Theorem
Assignment: Problem
Feb 5 Problems 1.4.18,19
Read 1.5 Borel Measures
M Feb 5 Lecture 8 Review of Problems 1.4.18,19 (no volunteers)
1.4 (part VI) The Uniqueness Theorem
1.5 Lebesgue Measure on the Real Line (part I)
Construction of Lebesgue measure
Assignment: Problems 1.4.20
Feb 12 Problems
5
W 7 Lecture 9 1.5 Lebesgue Measure on the Real Line (part I continued)
Finishing the construction
Assignment: Read 2.1 Measurable Functions
Feb 12 Problems 1.5.30,32
M 12 Lecture 10 Properties of Lebesgue measure and The Cantor Set
Assignment: Problems
Feb 19 Problems 1.5.27,28,29
6
W 14 Lecture 11 1.5 Lebesgue Measure on the Real Line (part II)
Lebesgue-Stieltjes measures
The Next Big Step: Integration
2.1 Measurable Functions and Integration
M 19 Lecture 12 2.1 Measurable Functions and Integration (continued)
Assignment: Problems 2.1.1,2,3
Feb 26 Problems
7
W 21 Lecture 13 2.2 Integration of (Nonnegative) Real Valued Functions
Monotone Convergence Theorem
Assignment: Problems 2.1.4,8,10
Feb 26
M 26 Lecture 14 2.2 (continued) Monotone Convergence Theorem
Applications of the MCT
Assignment: Problems 2.2.12,13,14
Mar 5
8
Exam 1 Problems 2,4,6 Due March 14
Remaining Problems Due March 26
W 28 Lecture 15 2.2 (continued) Fatou's Lemma
2.3 Integrals of more general functions, L^1
Lebesgue Dominated Convergence Theorem
Assignment: Problems 2.3.19
Mar 5
M Mar 5 Sick
W Mar 7 Lecture 16 2.3 (continued) The Reimann Integration Theorem
2.3 Part II Complex valued functions
Assignment: Problems 2.2.15,16
Mar 12 Problems 2.3.18
9
M Mar 12 Lecture 17 Complex valued functions (continued)
2.5 Product Measures; Lebesgue Measure on R^n (pp 22 ff)
Assignment: Problems 2.5.46
Mar 26 Problems
10
W Mar 14 Lecture 18 2.5 Product Measures (continued)
2.4 Introduction to Measure Theoretic Convergence
Exam 1 Problems 2,4,6 Due Today
M Mar 19 Spring Break
W Mar 21 Spring Break
M Mar 26 Lecture 19 2.4 Convergence of Functions
Assignment: Problems 2.4.32-35,39
April 2 Exam reworks (if you didn't get full credit)
11
Exam 1 Remaining Problems Due
W 28 Lecture 20 2.5 Fubini's Theorem
Assignment: Problems 2.5.45,50
April 2
Note: What we have done in sections 1.5 and 2.5 is extended in section 2.6
to give Lebesgue measure on R^n. We may not cover this explicitly
in class, but it is definitely recommended reading.
M April 2 Lecture 21 3.1 Signed measures and measure decomposition
Assignment: Problems 3.1.1,2,6 (Also interesting are 3 and 4)
April 9
12
W 4 Lecture 22 3.2 Lebesgue decomposition and Radon-Nikodym derivatives
Assignment: Problems 3.2.8,9,13 (Also interesting 11)
April 9 Problems
M 9 Lecture 23 3.3-4 The Fundamental Theorem of Calculus
Bounded Variation and Absolute Continuity for Functions
Assignment: Problems 3.3.24-26; 3.4.27,30,,31,33,37 (also interesting: 42)
April 16
13
W 11 Lecture 24 3.3-4 (continued)
M 16 Lecture 25 Functional Analysis and L^p spaces (5.1,5 & 6.1)
Assignment: Problems 6.1.1-3
April 23 5.1.1,2,5,6 (Read Section 5.1)
14
W 18 Lecture 26 Functional Analysis and L^p spaces (continued)
Assignment: Problems 6.1.6,7,9
April 23 5.1.3,8
M 23 Lecture 27 Bounded Linear Functionals and Riesz Representation
(5.1,2; 6.2)
Assignment: Problems
April 30 5.1.9,10,11
15 6.2.18
W 25 Lecture 28 Riesz Representation for L^p (last lecture)
Exam 1 reworks for odds due
W May 2 3:00 PM Exam 2 (2,3,6,7) Due; (1,4,5 Extra Credit)