Syllabus for Math 103

This syllabus is tentative and will be updated irregularly. The homework page will be updated on the regular basis.

 

Math 103 will concentrate on Measure Theory and Lebesgue integration with the goal of helping graduate students to prepare for the Analysis certification exam. The exact syllabus will depend on the interests and backgrounds of the students enrolled. From there we will cover as much of Chapters 1-7 as time permits. Ideally the students in the class should have had undergraduate classes in abstract analysis.

 

 

 

Lectures

Sections in Text

Brief Description

Day 1,  9/26/07

Sections 1.1-1.2

Introduction, algebras, sigma-algebras, Borel sigma-algebras, product sigma-algebras

Day 2, 9/28/07

Sections 1.2-1.3

Product sigma-algebras on Rn , measures, their examples and properties, measure completion

Day 3, 10/1/07

Section 1.4

Outer measure and Caratheodory Theorem, premeasures

Day 4, 10/3/07

Sections 1.4-1.5

Premeasures and outer measures, algebra of h-intervals on R, premeasures on R

Day 5, 10/4/07

x-hour

Section 1.5

Premeasures on R, Lebesgue-Stieltjes measure, Lebesgue-Stieltjes measurable sets and their relations to Gδ and Fσ  sets

Day 6, 10/5/07

Section 1.5

Lebesgue measure and Cantor set

Day 7, 10/8/07

Section 2.1

Measurable Functions and their properties. Polar decomposition and simple functions

Day 8, 10/10/07

Section 2.1 and start Section 2.2

Positive functions as limits of simple functions. Measurable functions that are a.e. equal. Integration of nonnegative functions

Day 9, 10/12/07

Section 2.2

Monotone Convergence Theorem and its Corollories

Day 10, 10/15/07

Section 2.3

Integration of complex functions and Dominated Convergence Theorem

Day 11, 10/17/07

Final day for electing use of the Non-Recording option.

Section 2.3

Integration of series, approximation of L1   functions by simple functions, taking limits and derivatives under the integral sign, Riemann integral

Day 12, 10/18/07

x-hour instead of the class on 10/19/07

Section 2.4

Modes of convergence and convergence in measure

10/19/07

Homecoming weekend

No class

 

 

Day 13, 10/22/07

Section 2.4 and start Section 2.5

Egoroff Theorem, Lusin Theorem, rectangles

Day 14, 10/24/07

The takehome Midterm

will be given out on this day and it will be due Monday October 29

Section 2.5

Monotone Class Lemma, product measures

Day 15, 10/26/07

Finish Section 2.5 and start Section 3.1

Fubini-Tonelli Theorem and isgned measures

Day 16, 10/29/07

Section 3.1

Hahn and Jordan decomposition Theorems

Day 17, 10/31/07

Finish Section 3.1 and start Section 3.2

Finish Jordan decomposition Theorem, absolute continuity of one measure with respect to another, prepare for the Lebesgue-Radon-Nikdym Theorem

Day 18, 11/2/07

Section 3.2

Lebesgue-Radon-Nikodym Theorem and Radon-Nikodym derivatives

Day 19, 11/5/07

Start Section 5.1

Norms, product norm, quotient norm

Day 20, 11/7/07

Section 5.1

Continuous operators, operator norm, Banach algebras

Day 21, 11/9/07

Section 5.2

Linear functionals, Hahn-Banach Theorem

Day 22, 11/12/07

Section 5.2 and start Section 5.3

Applications of Hahn Banach Theorem, Baire Category Theorem

Day 23, 11/14/07

11/15/07 is the final day to withdraw from the course

Section 5.3

Open Mapping and Closed Graph Theorems

Day 24, 11/15/07

x-hour

Section 5.3 and start Section 5.5

Uniform Boundedness Theorem, inner products, Schwarz inequality

Day 25, 11/16/07

Section 5.5

Hilbert spaces, Parallelogram Law, Pythagorean Theorem

Day 26, 11/19/07

Section 5.5

Orthogonal decomposition with respect to a closed subspace of a Hilbert space

5:50 PM 11/20/07-

7:45 AM 11/26/07 Thanksgiving Recess

 

 

 

Day 27, 11/26/07

Section 5.5

Bessel Inequality, Parseval Identity, existence of an orthonormal basis, separable Hilbert spaces

Day 28, 11/28/07

Section 6.1

Lp  spaces, Holder and Minkowski inequalities

Day 29, 11/29/07

x-hour

Oral Presentation of Homework Problems

 

Day 30 11/30/07

Oral Presentation of Homework Problems

 

Day 31, 12/3/07

Wrap Up

Lp  spaces as Banach spaces, relations between Lp  Lq  and Lr spaces

The takehome final exam will be distributed on 12/7/07 and it will be due

12/11/07