COURSE: MATH 5210, CALL #12436
TIME: 9:45-11:05 TR, PLACE: Room 313 of Gilbreath Hall
INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308G of Gilbreath Hall
OFFICE HOURS: 10:15-11:10 MWF, 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 Academic Press. We will use a preliminary version of the book which will be available in PDF and PostScript formats on the internet. Since the book is a "work in progress," revisions will be made throughout the term. You can get access to the latest version(s) at:
SUPPLEMENTAL 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 Royden 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:
TENTATIVE OUTLINE:
| Chapter 1: Fundamentals | Chapter 2: The Theory of Measure |
| 1.1 Elementary Set Theory | 2.0 Introduction |
| 1.2 Relations and Orderings | 2.1 Semi-Rings and Algebras of Sets |
| 1.3 Cardinality and Countability | 2.2 Measure |
| 1.4 The Topology of Rn | 2.3 Outer Measures and Measurable Sets |
| 1.5 Real-Valued Functions | 2.4 Lebesgue Measure |
| 1.6 The Stone-Weierstrass Theorem | 2.5 Borel Sets and Lebesgue Measurable Sets |
| 1.7 The Riemann Integral | 2.6 Measurable Functions |
| | 2.7 Convergence in Measure |
| Chapter 3: The Lebesgue Integral | Chapter 4: Special Topics in Integral |
| 3.1 Upper Functions | and Applications |
| 3.2 Lebesgue Integrable Functions | 4.1 Basic Concepts |
| 3.3 Riemann Integral and Lebesgue Integral | 4.2 Mathematical Model for Probability |
| 3.4 Convergence and Approximation | 4.3 Random Variables and Distributions |
| 3.5 Product Measure and Fubini's Theorem | 4.4 Expectation |
| 3.6 Differentiation and Bounded Variation Functions | 4.5 The Law of Large Numbers |
| and Central Limit Theorems | |
IMPORTANT DATES:
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