# Difference between revisions of "Math 641: Functions of a Real Variable"

### Title

Functions of Real and Complex Variables 1.

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. Lp spaces
• Hölder's Inequality
• Minkowski's Inequality
• Completeness of Lp
• Density of Cc in Lp
4. Measures on product spaces
• Tonelli Theorem
• Fubini Theorem
5. Measures on topological spaces
• Borel σ-algebra
• Locally compact Hausdorff spaces
• Urysohn's Lemma
• Partitions of unity
• Borel measures
• Locally finite measures
• Regular measures
• Riesz Representation Theorem (for positive linear functionals on Cc)
• Convergence in measure
• Hahn Decomposition Theorem
• Jordan Decomposition Theorem
• Mutually singular measures
• Lebesgue Decomposition Theorem
• Lusin's Theorem
• Egorov's Theorem
6. Lebesgue measure on Rn
• Mapping properties of Lebesgue measure
• 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