REAL ANALYSIS 1 - Fall 2006

Henri Lebesgue, 1875-1941

COURSE: MATH 5210-001

TIME: 4:40-6:00 MW, PLACE: Room 476 of Brown Hall

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

OFFICE HOURS: 9:00-10:00 TR, PHONE: 439-6979 (Math Office 439-4349)

E-MAIL: gardnerr@etsu.edu
WEBPAGE: http://www.etsu.edu/math/gardner/gardner.htm

TEXT: Real Analysis, Third Edition, by H. L. Royden.

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 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. In addition, we will have a number of "convergence theorems" related to the Lebesgue integral, which are not true in the setting of Riemann integration.

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.

A NOTE ABOUT HOMEWORK: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Several of the homework problems are fairly standard for this class, and you may find proofs online. However, the online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class.

TENTATIVE OUTLINE:
Chapter 1: Set Theory (Sections 4, 5, and 6).
algebra of sets, σ-algebras, axiom of choice, countable sets, rational numbers.
Chapter 2: The Real Number System (Sections 1,2 3, and 7).
axioms of the real numbers, completeness, natural numbers, rational numbers, extended real numbers, Borel sets.
Chapter 3: Lebesgue Measure.
outer measure, measurable sets, Lebesgue measure, measurable functions, characteristic functions, Littlewood's principles, Egoroff's Theorem, Lusin's Theorem.
Chapter 4: The Lebesgue Integral.
Riemann integral, step functions, simple functions, Lebesgue integral of a bounded function, Bounded Convergence Theorem, Fatou's Lemma, Monotone Convergence Theorem, Lebesgue Convergence Theorem, general Lebesgue Integral, Convergence in measure.
Chapter 5: Differentiation and Integration.
monotone functions, bounded variation, differentiation of an integral, absolute continuity, convex functions, Jensen's Inequality.

IMPORTANT DATES:

Interesting Questions Which Were Raised
  1. From Mr. Mullins: "What is the cardinality of the collection of Borel sets?" Any opinions, feedback, or references?
    Contrary to my in-class speculations, according Corollary 4.5.3 of Inder Rana's An Introduction to Measure and Integration (2nd Edition, A,M.S. Graduate Studies in Mathematics Volume 45, 2002) the cardinality is c, the cardinality of the continuum. (Do you see why I view this as bad news?) The proof uses transfinite induction (ouch!).
  2. From Dr. Bob: "Why 'Fσ' and 'Gδ'?"
    Ms. Harrell found the following two references (But BEWARE! - They are from Wikipedia!): Fσ sets and Gδ sets. These sites imply that the F and G are from French and the σ and δ are from German.
  3. From Dr. Bob: "How many (Lebesgue) nonmeasurable sets are there?" Any reliable references?
    Mr. Peyton found a website that shows that (assuming the continuum hypothesis) there are aleph_2 (the cardinality of the power set of the reals) nonmeasurable sets. One way to illustrate this is to take each element of the power set of a translation of the Cantor set (C+1; denote this power set P(C+1)) and union it with the nonmeasurable subset P of [0,1) which we created in class: N = {X ∪ P | X∈ P(C+1)}. Since the cardinality of C is aleph_1 (the cardinality of the continuum), then N has cardinality aleph_2. Each element of N is nonmeaurable since it is a disjoint union of nonmeasurable P and a measurable subset of C (recall that C has measure 0).

Homework
Section
Problems
Due Date
Points
1.4
19b
WED 9/6
3
1.6
23, 25
WED 9/13
3+3=6
2.1, 2.7
5b, 53
WED 9/20
3+6=9
3.1
1, 3
WED 9/27
3+3=6
3.2
6, 7
WED 10/4
3+3=6
3.3
9, 11, 14b
WED 10/11
3+2+4=9
3.4
16, 17a
MON 10/23
3+3=6
3.5
19, 21a, 21b, Bonus: 26
MON 11/6
3+3+3+(16)=9+(16)
4.3
3, 6, Bonus: 5
MON 11/20
3+3+(3)=6+(3)
4.4
10a, 14b, Bonus: one from 4.5
WED 11/29
5+3+(3)=8+(3)
5.1, 5.2
3a (replace "maximum" with "minimum"), 3b, 7a, 10a, 10b, Bonus: 7b
WED 12/6
3+2+3+3+3+(3)=14+(3)
TOTAL
-
-
82+(25)
The numbers in parentheses represent bonus problems.

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