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
   • 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
   • Monotone Convergence Theorem
   • Fatou's Lemma
   • Dominated Convergence Theorem
   • Effect of sets of measure zero
3. Product Measures
4. Differentiation and integration
5. Miscellaneous
   • Borel sets