Measure and Integration

Calendar

LEC #	TOPICS	KEY DATES
1	Why Measure Theory? Measure Spaces and Sigma-algebras Operations on Measurable Functions (Sums, Products, Composition) Borel Sets
2	Real-valued 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 Non-negative Measurable Functions Interchanging Summation and Integration Fatou’s Lemma
5	Integral of Complex Functions Dominated Convergence Theorem Sets of Measure Zero Completion of a Sigma-algebra
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 Sigma-algebra, 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 Non-measurable 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) Vitali-Caratheodory 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 Riesz-Fischer 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^p-norm Smooth Functions Dense in L^p
18	Fubini’s Theorem in R^n for Non-negative 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 (Hardy-Littlewood 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