| LEC # | TOPICS | KEY DATES |
|---|---|---|
| 1 | Why Measure Theory? Measure Spaces and Sigma-algebras Operations on Measurable Functions (Sums, Products, Composition) Borel Sets |
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| 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 |
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| 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 |
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| 5 | Integral of Complex Functions Dominated Convergence Theorem Sets of Measure Zero Completion of a Sigma-algebra |
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| 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 |
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| 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 |
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| 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 |
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| 10 | Integration as a Linear Functional Riesz Representation Theorem for Positive Linear Functionals Lebesgue Integral is the "Completion" of the Riemann Integral |
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| 11 | Lusin's Theorem (Measurable Functions are nearly continuous) Vitali-Caratheodory Theorem |
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| 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 |
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| 14 | Convex Functions Jensen's Inequality Hölder and Minkowski Inequalities |
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| 15 | L^p Spaces, 1 Leq p Leq Infty Normed Spaces, Banach Spaces Riesz-Fischer Theorem (L^p is complete) |
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| 16 | C_c Dense in L^p, 1 Leq p < Infty C_c Dense in C_o (Functions which vanish at Infty) |
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| 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 |
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| 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 |
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| 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 |
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| 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) |
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| 23 | Lebesgue's Differentiation Theorem The Lebesue Set of an L^1 Function Fundamental Theorem of Calculus I |
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| 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 |