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 |
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Day 1, Wednesday 9/23/09 |
Section 1.1 and the beginning of Section 1.2 |
What does one want from a measure? Some arising problems. Sigma algebras. |
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Day 2, Friday 9/25/09 |
Section 1.2 and the beginning of Section 1.3 |
Examples of sigma algebras, Borel sigma algebras, product sigma algebras. |
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Day 3, Monday 9/28/09 |
Sections 1.2 and 1.3 |
Measures, various types of measure. |
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Day 4, Tuesday 9/29/09 x-hour |
Section 1.3 and the beginning of Section 1.4 |
Complete measures, completion. Outer measure. |
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Day 5, Wednesday 9/30/09 |
Section 1.4 |
Outer measureable sets. Caratheodory’s Theorem. |
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Day 6, Friday 10/2/09 |
Section 1.4 and start reading Section 1.5 |
Premeasure. , extension of a premeasured. Borel measure. |
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Day 7, Monday 10/5/09 |
Read Section 1.5 |
Premeasure coming from an increasing right continuous function |
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Day 8, Wednesday 10/7/09 Final day for electing use of the Non-Recording option. |
Read Section 1.5 |
Lebesgue-Stieltjes measure |
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Day 9, Friday 10/9/09 |
Read Section 1.5 and start reading Section 2.1 |
Lebesgue measurable sets. Cantor set. |
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Day 10, Monday 10/12/09 |
Read Section 2.1 |
Measurable functions and operations on them. |
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Day 11, Wednesday 10/14/09 |
Read Section 2.1 |
Simple functions and representing measurable functions as their limits. |
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Day 12, Friday 10/16/09 |
Read Section 2.2 |
Integration of nonnegative functions. Monotone Convergence Theorem. |
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Day 13, Monday 10/19/09 |
Read Section 2.2 and start reading Section 2.3 |
Integration of series of nonnegative measurable functions. Nonnegative function that is almost everywhere zero. Fatou’s Lemma. |
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Day 14, Wednesday 10/21/09 |
Read Section 2.3 |
The vector space of integrable functions. | ∫f| ≤ ∫|f| |
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Friday 10/23/09 Homecoming weekend No class, instead we shall have an x-hour on Tuesday 10/27/09 |
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Day 15, Monday 10/26/09 |
Read Section 2.3 |
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Day 16, Tuesday 10/27/09 x-hour instead of a class on Friday 10/23/09 |
Read Section 2.3 |
Comparison of the Lebesgue and the Riemann integral. |
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Day 17, Wednesday 10/28/09 The take home Midterm
Exam will be distributed on this day It will be due on Monday 11/2/09 |
Read Section 2.4 |
Modes of Convergence. Convergence in measure, L1 convergence, a.e. convergence. |
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Friday 10/30/09 No class, instead we shall have an x-hour on Tuesday 11/3/09 |
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Day 18, Monday 11/2/09 The Midterm Exam
is due Note that Tuesday 11/03/09 is the Final day for dropping a fourth
course without a grade notification of “W” |
Read Section 2.4 |
Egoroff’s Theorem and Lousin Theorem |
|
Day 19, Wednesday 11/4/09 |
Read Section 2.5 |
Product Measure. Monotone Class Lemma |
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Day 20, Friday 11/6/09 |
Read Section 2.5 |
The Fubinni-Tonneli Theorem. |
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Day 21, Monday 11/9/09 |
Section 3.1 |
Signed measure and the Hahn decomposition Theorem |
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Day 22, Tuesday 11/10/09 x-hour instead of the class on Friday 10/30/09 |
Section 3.1 and start Section 3.2 |
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Day 23, Wednesday 11/11/09 |
Section 3.2 |
The Lebesgue-Radon-Nikodym Theorem. |
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Day 24, Friday 11/13/09 Final day to withdraw from a course |
Section 3.2 |
Radon-Nikodym derivative. |
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Day 25, Monday 11/16/09 |
Section 5.1 |
Normed Vector spaces. Operator norm. Continuous Linear maps. |
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Day 26, Wednesday 11/18/09 |
Section 6.1 |
Lp spaces. Holder inequality. Minkowski inequality. |
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Day 27, Friday 11/20/09 |
Section 6.1 |
Lp is a Banach space. Simple functions are dense in Lp . Relations between Lp spaces for different p. |
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Day 28, Monday 11/23/09 |
Dual Spaces, parts of Sections
5.2 and 6.2 |
Notion of a dual space. Dual of Lp and conjugate exponents. |
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Thanksgiving Recess 5:50 PM on Tuesday 11/24/09 – 7:45 AM on Monday 11/30/09 No Class |
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Day 29 Monday 11/30/09 |
Oral Homework Presentations will be done on this day |
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Tuesday 12/1/09 x-hour if necessary |
Oral Homework Presentations will be done on this day |
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Day 30, Wednesday 12/2/09 The takehome Final Exam will be
distributed on this day and it will be due on Tuesday December 8 at Noon |
Loose ends. |
Loose ends. |
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Pre-Examination Break Thursday 12/3/09-Friday 12/4/09 |
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Tuesday 12/8/09 The Final Exam is
due at Noon |
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