Real Analysis 1 - Fall 2004

COURSE: MATH 5210, CALL #33441

TIME: 9:45-11:05 TR, PLACE: Room 122 of Rogers-Stout Hall

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308G of Gilbreath Hall

OFFICE HOURS: 1:00-2:00 T, 10:25-11:20 W PHONE: 439-6977 (Math Office 439-4349)

TEXT: The primary text is Real Analysis with an Introduction to Wavelets and Applications. This text is being authored by Don Hong (ETSU), Jianzhong Wang (Sam Houston State University), and Bob Gardner (ETSU). It is currently under contract with Elsevier Press/Academic Press. We will use a preliminary version of the book which is available in PDF form on the internet. You can access to the text (in part) at (to be removed from the server by September 10):

http://www.etsu.edu/math/gardner/5210/analysis-text.pdf
You can view the webpage of the book here.

ABOUT THE COURSE: This class offers a standard introduction to the theory of functions of a real variable from the measure theoretic perspective. As commented on page 1 of the Royden's Real Analysis text, we will cover "a portion of the material that every graduate student in mathematics must know." Whereas the undergraduate real analysis class presents the results of calculus from a rigorous perspective, we will introduce fundamentally new ideas which are basic extensions of the results from calculus. In particular, we will put a weight or "measure" on certain sets of real numbers. This measure will be used to define a new type of integral called the Lebesgue integral. Recall that a function is Riemann integrable if and only if it is discontinuous on a "small" set (namely, a set of measure zero). The Lebesgue integral is much more flexible and will allow us to integrate a much larger class of functions.

GRADING: We will have two tests (T1 and T2) and homework (HW) will be taken up a regular intervals (weekly). Your average will be computed as follows:

AVERAGE = (T1 + T2 + 2HW)/4.
Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate.

TENTATIVE OUTLINE:

Chapter 1: Fundamentals
Chapter 2: The Theory of Measure
1.1 Elementary Set Theory
2.1 Classes of Sets
1.2 Relations and Orderings
2.2 Measures on Rings
1.3 Cardinality and Countability
2.3 Outer Measure and Lebesgue Measure
1.4 The Topology of Rn
2.4 Measurable Functions
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2.5 Convergence of Measurable Functions
Chapter 3: The Lebesgue Integral
Chapter 4: Special Topics of Integration
3.1Riemann Integral and Lebesgue Integral
4.1 Differentiation and Integration
3.2 The General Lebesgue Integral
4.2 Mathematical Models for Probability
3.3 Convergence and Approximation
4.3 Convergence and Limit Theorems
3.4 Lebesgue Integrals in the Plane
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IMPORTANT DATES:
Monday, September 6 = Labor Day Holiday - University closed.
Friday, September 10 = Last day for 75% refund.
Monday, September 27 = Last day to drop without a grade of "W."
Monday and Tuesday, October 18 and 19 = Fall Break - No classes.
Monday, October 25 = Last day to drop without dean's permission.
Thursday, October 28 = Test 1 covering Chapter 1 and parts of Chapter 2.
Thursday and Friday, November 25 and 26 = Thanksgiving Holiday - University closed.
Wednesday, December 8 = Last day to withdraw from the university.
Friday, December 10 = Last day of class.
Thursday, December 16 = Test 2, 8:00 a.m. to 10:00 a.m.

Homework
Section
Problems
Due Date
Points
1.1
3, 5b, 6a, 6b, 10 (choose 3)
Tuesday 9/7
3+3+3=9
1.2
5a, 5b, 16, 18, 21 (choose 3)
Thursday 9/16
3+3+3=9
1.3
1, 9, 10, 13 (choose 2), BONUS: 11, 14, 15
Thursday 9/23
3+3=6
1.4
1, 2, 6, 7, 11 16 (choose 3)
Thursday 9/30
3+3+3=9
1.4
17, 18, 20, 21, 22, 24 (choose 3)
Thursday 10/7
3+3+3=9
2.1
1c, 2a, 2b, BONUS: 3c
Friday 10/15
3+3+3=9
2.1
4, 9, BONUS: 3c (again), 10a
Tuesday 10/26
3+3=6
2.3
3, 9, BONUS: 5
Thursday 11/9
3+3=6

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