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

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. L^p spaces
   • Hölder's Inequality
   • Minkowski's Inequality
   • Completeness of L^p
4. Measures on product spaces
   • Tonelli Theorem
   • Fubini Theorem
5. Differentiation on R and integration
   • Derivative of integral is the integrand a.e.
   • Functions of bounded variation
   • Absolutely continuous functions
   • Integrating derivatives of absolutely continuous functions
6. Measures on topological spaces
   • Borel sets
   • Convergence in measure
   • Hahn Decomposition Theorem
   • Jordan Decomposition Theorem
   • Radon-Nikodym Theorem
   • Riesz Representation Theorem (for bounded linear functions on L^p)
   • Mutually singular measures
   • Lebesgue Decomposition Theorem
   • Lebesgue measure
     • Mapping properties of Lebesgue measure
   • Lusin's Theorem
   • Egorov's Theorem