The lecture notes were prepared in LaTeX by Ethan Brown, a former student in the class. He used Professor Viaclovsky’s handwritten notes in producing them.
| LEC # | TOPICS | LECTURE NOTES |
|---|---|---|
| 1 |
Why Measure Theory?
Measure Spaces and Sigma-algebras Operations on Measurable Functions (Sums, Products, Composition) Borel Sets |
(PDF) |
| 2 |
Real-valued Measurable Functions
Limits of Measurable Functions Simple Functions Positive Measures Definition of Lebesgue Integral |
(PDF) |
| 3 |
Riemann Integral
Riemann Integrable <-> Continuous Almost Everywhere Comparison of Lebesgue and Riemann Integrals Properties of Positive Measures Elementary Properties of the Lebesgue Integral |
(PDF) |
| 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 |
(PDF) |
| 5 |
Integral of Complex Functions
Dominated Convergence Theorem Sets of Measure Zero Completion of a Sigma-algebra |
(PDF) |
| 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 |
(PDF) |
| 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 |
(PDF) |
| 8 |
Caratheodory Criterion
Cantor Set There exist (many) Lebesgue measurable sets which are not Borel measurable |
(PDF) |
| 9 |
Invariance of Lebesgue Measure under Translations and Dilations
A Non-measurable Set Invariance under Rotations |
(PDF) |
| 10 |
Integration as a Linear Functional
Riesz Representation Theorem for Positive Linear Functionals Lebesgue Integral is the “Completion” of the Riemann Integral |
(PDF) |
| 11 |
Lusin’s Theorem (Measurable Functions are nearly continuous)
Vitali-Caratheodory Theorem |
(PDF) |
| 12 |
Approximation of Measurable Functions by Continuous Functions
Convergence Almost Everywhere Integral Convergence Theorems Valid for Almost Everywhere Convergence Countable Additivity of the Integral |
(PDF) |
| 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 |
(PDF) |
| 14 |
Convex Functions
Jensen’s Inequality Hölder and Minkowski Inequalities |
(PDF) |
| 15 |
L^p Spaces, 1 Leq p Leq Infty
Normed Spaces, Banach Spaces Riesz-Fischer Theorem (L^p is complete) |
(PDF) |
| 16 |
C_c Dense in L^p, 1 Leq p < Infty
C_c Dense in C_o (Functions which vanish at Infty) |
(PDF) |
| 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 |
(PDF) |
| 18 | Fubini’s Theorem in R^n for Non-negative Functions | (PDF) |
| 19 |
Fubini’s Theorem in R^n for L^1 Functions
The Product Measure for Products of General Measure Spaces |
(PDF) |
| 20 |
Fubini’s Theorem for Product Measure
Completion of Product Measures Convolutions |
(PDF) |
| 21 |
Young’s Inequality
Mollifiers C^{Infty} Dense in L^p, 1 Leq p < Infty |
(PDF) |
| 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) |
(PDF) |
| 23 |
Lebesgue’s Differentiation Theorem
The Lebesue Set of an L^1 Function Fundamental Theorem of Calculus I |
(PDF) |
| 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 |
(PDF) |
