Math 641: Functions of a Real Variable

Catalog Information

Title

Functions of Real and Complex Variables 1.

Credit Hours

3

Prerequisite

Math 542 or instructor's consent

Description

Fundamentals of measure and integration, Borel measures, product measures, L^ spaces, introduction to functional analysis, Radon Nikodym theorem, differentiation theory, Fourier transforms.

Desired Learning Outcomes

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.

1. Abstract measure theory
   • σ-algebras
   • Measures
     • Positive measures
     • Signed measures
     • σ-finite measures
     • Complete measures
   • Measurable spaces
   • Measure spaces
2. Abstract integration theory
   • Abstract measurable mappings
   • Measurable real- and extended-real-valued functions
   • Integrating simple functions
   • Integrating nonnegative functions
   • Integrating L¹ 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
3. Operations on measures
   • Absolutely continuous measures
   • Mutually singular measures
   • Lebesgue Decomposition Theorem
   • Radon-Nikodym Theorem
   • Hahn Decomposition Theorem
   • Jordan Decomposition Theorem
4. 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
5. Convergence results
   • Types of convergence
     • Convergence in L^p-norm
     • Almost-everywhere convergence
     • Almost-uniform convergence
     • Convergence in measure
   • Relationships between different types of convergence
     • Egoroff's Theorem
6. Measures on abstract product spaces
   • Existence of product measure
   • Tonelli's Theorem
   • Fubini's Theorem
7. Measures on topological spaces
   • Borel σ-algebra
   • 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
8. Lebesgue measure on Rⁿ
   • 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