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

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#* Dominated Convergence Theorem | #* Dominated Convergence Theorem | ||

#* Effect of sets of measure zero | #* Effect of sets of measure zero | ||

+ | # L<sup>p</sup> spaces | ||

+ | #* Hölder's Inequality | ||

+ | #* Minkowski's Inequality | ||

+ | #* Completeness of L<sup>p</sup> | ||

# Product Measures | # Product Measures | ||

# Differentiation and integration | # Differentiation and integration |

## Revision as of 16:26, 13 August 2008

## 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

- 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 - Monotone Convergence Theorem
- Fatou's Lemma
- Dominated Convergence Theorem
- Effect of sets of measure zero

- L
^{p}spaces- Hölder's Inequality
- Minkowski's Inequality
- Completeness of L
^{p}

- Product Measures
- Differentiation and integration
- Miscellaneous
- Borel sets