# Math 641: Functions of a Real Variable

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## Contents

## 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

### Prerequisites

### Minimal learning outcomes

- Abstract measure theory
- σ-algebras
- Measures
- Positive measures
- Signed measures
- σ-finite measures
- Complete measures

- Measurable spaces
- Measure spaces

- Abstract integration theory
- Abstract measurable mappings
- Measurable real- and extended-real-valued functions
- Integrating simple functions
- Integrating nonnegative functions
- Integrating L
^{1}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
- Absolutely continuous measures
- Mutually singular measures
- Lebesgue Decomposition Theorem
- Radon-Nikodym Theorem
- Hahn Decomposition Theorem
- Jordan Decomposition Theorem

- L
^{p}spaces- Hölder's Inequality
- Minkowski's Inequality
- Completeness of L
^{p} - Density of C
_{c}in L^{p} - Convergence in norm
- Almost-everywhere convergence
- Almost-uniform convergence
- Convergence in measure
- Egoroff's Theorem

- Measures on product spaces
- Tonelli Theorem
- Fubini Theorem

- 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

- Lebesgue measure on
**R**^{n}- Existence
- Composition with affine maps
- 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