Lecture 
Date 
What was
covered 
Notes 
Textbook Section 
1 
Aug. 29 
Numbers (N,Z,Q,R,C). Relations (equivalence, order) 
get assignment 1 
1.1, 1.3, 1.4 
2 
Aug. 31 
Metric spaces. Completion of rationals. 
 
1.2, 1.5 
3 
Sep. 5 
(Axiom of Choice and Zorn's Lemma. Countability.) Metric Spaces. Topology. Topological Spaces 
return assignment 1; get assignment 2 
1.2 (1.5, 1.6) 2.1 
4 
Sep. 7 
Completion of Metric Spaces 


5 
Sept.12 
Norms, Banach Spaces, Separability, Scalar products, Hilbert Spaces (I) 
return assignment 2; get assignment 3 
3.1 
6 
Sept.14 
Hilbert Spaces (II): orthonormality, Pythagora, Bessel, CauchySchwarz, nonseparability of l^inf 

3.2 
7 
Sept.19 
completion of l^p 
return assignment 3; get assignment 4 

8 
Sept.21 
Duality in Hilbert Spaces. Riesz Representation theorem 

3.2, 3.3 
9 
Sept.26 
RiemeannStieltjes Integral (1) 
return assignment 4; get assignment 5 
4.1 
10 
Sept.28 
Functions in BV space. Jordan decomposition. Pure point part of a BV function 

4.15 
11 
Oct. 3 
RiemannStieltjes integral (2) 
return assignment 5; get assignment 6 
4.1 
12 
Oct. 5 
Cantor sets. Cantor function. 

4.2 
13 
Oct.10 
Cantor Function (2) 
get assignment 7 
4.2 , MATH 3206 
14 
Oct.12 
Measurable spaces and functions. Borel and Baire Sigma algebras. 
return assignment 6 
4.3 (and Chapter 2 for Gdelta, Fsigma sets), MATH 3206 
15 
Oct.17 
L^1 as completion of C[a,b] 
return assignment 7 
4.4 
 
Oct.18 
REVIEW SESSION 
3:15pm4:50pm 
CSIC 4122 
16 
Oct.19 
MID TERM EXAM 


17 
Oct.24 
Lipschitz maps acting on L^1. Monotone Convergence Theorem (1). Partition of Unity 
get assignment 8 
4.4 
18 
Oct.26 
Measure of Open Sets. Additivity and Subadditivity. 

4.4 
19 
Oct.31 
RieszFischer Theorem. Luzin Theorem. 
return assignment 8; get assignment 9 
4.4 
20 
Nov. 2 
Luzin and Egorov theorems for L1 functions. Characterizations of abstract L1 space 

4.4 
21 
Nov. 7 
Monotone Convergence Theorem (2); Measure of Borel sets. 
return assignment 9; pick up assignment 10 
4.4 
22 
Nov. 9 
Regularity of Borel measures. 

4.4 and 4.5 (Theorem 4.5.6) 
23 
Nov.14 
Lebesgue and LebesgueStieljes integrals of L1 functions. Lp and Linf spaces. 
return assignment 10; get assignment 11 
4.4 
24 
Nov. 16 
Convergence Theorems: Fatou Lemma, Lebegues Dominated Convergence 

4.6 
25 
Nov.21 
BrezisLieb Theorem, Scheffe Lemma; Egorov Theorem 
return assignment 11; get assignment 12 
4.6 
 
Nov.23 
THANKSGIVING BREAK 

NO CLASS 
26 
Nov.28 
Product Measure. Fubini Theorem 

4.11 
27 
Nov.30 
Absolute continuity. Mutually singular measures. 
return assignment 12; get assignment 13 
4.7 
28 
Dec. 5 
VonNeumann theorem. Lebesgue decomposition theorem. RadonNikodym theorem. 

4.7 
29 
Dec. 7 
Absolute Continuity for measures and BV functions 
return assignment 13 
LAST CLASS 
 
Thurs., Dec.14 
REVIEW SESSION 
12:30pm2:00pm 
CSIC 4122 
 
Mon, Dec. 18 
FINAL EXAM 
CSIC 4122 
1:30pm3:30pm 