| 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
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| 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 | |