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. 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