LEC #  TOPICS  KEY DATES 

1 
Why Measure Theory?
Measure Spaces and Sigmaalgebras Operations on Measurable Functions (Sums, Products, Composition) Borel Sets 

2 
Realvalued Measurable Functions
Limits of Measurable Functions Simple Functions Positive Measures Definition of Lebesgue Integral 

3 
Riemann Integral
Riemann Integrable <> Continuous Almost Everywhere Comparison of Lebesgue and Riemann Integrals Properties of Positive Measures Elementary Properties of the Lebesgue Integral 

4 
Integral is Additive for Simple Functions
Monotone Convergence Theorem Integral is Additive for All Nonnegative Measurable Functions Interchanging Summation and Integration Fatou’s Lemma 

5 
Integral of Complex Functions
Dominated Convergence Theorem Sets of Measure Zero Completion of a Sigmaalgebra 

6 
Lebesgue Measure on R^n
Measure of Special Rectangles Measure of Special Polygons Measure of Open Sets (Approximate from within by Polygons) Measure of Compact Sets (Approximate from outside by Opens) Outer and Inner Measures 

7 
Definition of Lebesgue Measurable for Sets with Finite Outer Measure
Remove Restriction of Finite Outer Measure (R^n, L, Lambda) is a Measure Space, i.e., L is a Sigmaalgebra, and Lambda is a Measure 

8 
Caratheodory Criterion
Cantor Set There exist (many) Lebesgue measurable sets which are not Borel measurable 
Homework 1 due 
9 
Invariance of Lebesgue Measure under Translations and Dilations
A Nonmeasurable Set Invariance under Rotations 

10 
Integration as a Linear Functional
Riesz Representation Theorem for Positive Linear Functionals Lebesgue Integral is the “Completion” of the Riemann Integral 

11 
Lusin’s Theorem (Measurable Functions are nearly continuous)
VitaliCaratheodory Theorem 

12 
Approximation of Measurable Functions by Continuous Functions
Convergence Almost Everywhere Integral Convergence Theorems Valid for Almost Everywhere Convergence Countable Additivity of the Integral 
Homework 2 due 
13 
Egoroff’s Theorem (Pointwise Convergence is nearly uniform)
Convergence in Measure Converge Almost Everywhere > Converges in Measure Converge in Measure > Some Subsequence Converges Almost Everywhere Dominated Convergence Theorem Holds for Convergence in Measure 

14 
Convex Functions
Jensen’s Inequality Hölder and Minkowski Inequalities 

15 
L^p Spaces, 1 Leq p Leq Infty
Normed Spaces, Banach Spaces RieszFischer Theorem (L^p is complete) 

16 
C_c Dense in L^p, 1 Leq p < Infty
C_c Dense in C_o (Functions which vanish at Infty) 

17 
Inclusions between L^p Spaces? l^p Spaces?
Local L^p Spaces Convexity Properties of L^pnorm Smooth Functions Dense in L^p 

18  Fubini’s Theorem in R^n for Nonnegative Functions  
19 
Fubini’s Theorem in R^n for L^1 Functions
The Product Measure for Products of General Measure Spaces 

20 
Fubini’s Theorem for Product Measure
Completion of Product Measures Convolutions 
Homework 3 due 
21 
Young’s Inequality
Mollifiers C^{Infty} Dense in L^p, 1 Leq p < Infty 

22 
Fundamental Theorem of Calculus for Lebesgue Integral
Vitali Covering Theorem Maximal Function f in L^1 > Mf in Weak L^1 (HardyLittlewood Theorem) 

23 
Lebesgue’s Differentiation Theorem
The Lebesue Set of an L^1 Function Fundamental Theorem of Calculus I 

24 
Generalized Minkowski Inequality
Another Proof of Young’s Inequality Distribution Functions Marcinkiewicz Interpolation: Maximal Operator Maps L^p to L^p for 1 < p Leq Infty 
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