18.125 | Fall 2003 | Graduate

# 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

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

Fall 2003
Lecture Notes
Problem Sets