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 