18.125 | Fall 2003 | Graduate

Measure and Integration

Lecture Notes

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.

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

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

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 (PDF)
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


21 Young’s Inequality


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


Course Info

As Taught In
Fall 2003
Learning Resource Types
Lecture Notes
Problem Sets