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Revision as of 15:44, 3 April 2013
Contents
Catalog Information
Title
Functions of a Real Variable.
Credit Hours
3
Prerequisite
Math 541 or instructor's consent
Description
Abstract measure and integration theory; L(p)(?) spaces; measures on topological and Euclidean spaces.
Desired Learning Outcomes
Math 641 is a course in abstract measure and integration theory. There will be some repetition of topics between Math 541 and Math 641, but it is felt that the repetition will help solidify student understanding, and there will be a difference in approach, with the lowerlevel course taking a concrete approach restricted to Lebesgue measure.
Prerequisites
We strongly recommend that students take Math 541 prior to Math 641.
Minimal learning outcomes
Outlined below are topics that all successful Math 641 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.
 Abstract measure theory
 σalgebras
 Measures
 Positive measures
 Signed measures
 σfinite measures
 Complete measures
 Measurable spaces
 Measure spaces
 Abstract integration theory
 Abstract measurable mappings
 Measurable real and extendedrealvalued functions
 Integrating simple functions
 Integrating nonnegative functions
 Integrating L^{1} functions
 Integration on a measurable set
 Measures defined through integration
 Absolute continuity of integration
 Linearity of integration
 Monotone Convergence Theorem
 Fatou's Lemma
 Dominated Convergence Theorem
 Effect of sets of measure zero
 Operations on measures
 Absolutely continuous measures
 Mutually singular measures
 Lebesgue Decomposition Theorem
 RadonNikodym Theorem
 Hahn Decomposition Theorem
 Jordan Decomposition Theorem
 L^{p} spaces
 Hölder's Inequality
 Minkowski's Inequality
 Completeness of L^{p}
 Density of C_{c} in L^{p}
 Inclusion of L^{p} spaces
 Duality of L^{p} spaces
 Convergence results
 Types of convergence
 Convergence in L^{p}norm
 Almosteverywhere convergence
 Almostuniform convergence
 Convergence in measure
 Relationships between different types of convergence
 Egoroff's Theorem
 Types of convergence
 Measures on abstract product spaces
 Existence of product measure
 Tonelli's Theorem
 Fubini's Theorem
 Measures on topological spaces
 Borel σalgebras
 Locally compact Hausdorff spaces
 Urysohn's Lemma
 Partitions of unity
 Borel measures
 Locally finite measures
 Regular measures
 Radon measures
 Riesz Representation Theorem (for positive linear functionals on C_{c})
 Lusin's Theorem
 Lebesgue measure on R^{n}
 Existence
 Composition with affine maps
 Change of variable formula for integration
 Differentiation and integration on R
 Derivative of integral is the integrand a.e.
 Functions of bounded variation
 Absolutely continuous functions
 Integrating derivatives of absolutely continuous functions
Textbooks
Possible textbooks for this course include (but are not limited to):
Additional topics
Courses for which this course is prerequisite
Math 641 is not a prerequisite for any course.