# Difference between revisions of "Math 541: Real Analysis"

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− | + | # Lebesgue measure on <b>R</b><sup>n</sup> | |

+ | #* Inner and outer measures | ||

+ | #* Construction of Lebesgue measure | ||

+ | #* Properties of Lebesgue measure | ||

+ | #** Effect of basic set operations | ||

+ | #** Limiting properties | ||

+ | #** Its domain | ||

+ | #** Approximation properties | ||

+ | #** Sets of outer measure zero | ||

+ | #** Invariance w.r.t. isometries | ||

+ | #** Effect of dilations | ||

+ | #* Existence of nonmeasurable sets | ||

+ | # Lebesgue integration on <b>R</b><sup>n</sup> | ||

+ | #* Measurable functions | ||

+ | #* Simple functions | ||

+ | #* Approximation of measurable functions with simple functions | ||

+ | #* The extended reals | ||

+ | #* Integrating nonnegative functions | ||

+ | #* Integrating absolutely-integrable functions | ||

+ | #* Integrating on measurable sets | ||

+ | #* Basic properties of the Lebesgue integral | ||

+ | #** Linearity | ||

+ | #** Monotonicity | ||

+ | #** Effects of sets of measure zero | ||

+ | #** Absolute continuty of integration | ||

+ | #** Fatou's Lemma | ||

+ | #** Monotone Convergence Theorem | ||

+ | #** Dominated Convergence Theorem | ||

+ | #** Differentiation w.r.t. a parameter | ||

+ | #** Linear changes of variable | ||

+ | #** Compatibility with Riemann integration | ||

+ | # Fubini's Theorem for <b>R</b><sup>n</sup> | ||

+ | # L<sup>1</sup>, L<sup>2</sup>, and L<sup>∞</sup> | ||

+ | #* Their completeness | ||

+ | #* Approximation by smooth functions | ||

</div> | </div> | ||

## Revision as of 09:42, 13 August 2008

## Contents

## Catalog Information

### Title

Real Analysis.

### Credit Hours

3

### Prerequisite

### Description

Rigorous treatment of differentiation and integration theory; Lebesque measure; Banach spaces.

## Desired Learning Outcomes

### Prerequisites

### Minimal learning outcomes

- Lebesgue measure on
**R**^{n}- Inner and outer measures
- Construction of Lebesgue measure
- Properties of Lebesgue measure
- Effect of basic set operations
- Limiting properties
- Its domain
- Approximation properties
- Sets of outer measure zero
- Invariance w.r.t. isometries
- Effect of dilations

- Existence of nonmeasurable sets

- Lebesgue integration on
**R**^{n}- Measurable functions
- Simple functions
- Approximation of measurable functions with simple functions
- The extended reals
- Integrating nonnegative functions
- Integrating absolutely-integrable functions
- Integrating on measurable sets
- Basic properties of the Lebesgue integral
- Linearity
- Monotonicity
- Effects of sets of measure zero
- Absolute continuty of integration
- Fatou's Lemma
- Monotone Convergence Theorem
- Dominated Convergence Theorem
- Differentiation w.r.t. a parameter
- Linear changes of variable
- Compatibility with Riemann integration

- Fubini's Theorem for
**R**^{n} - L
^{1}, L^{2}, and L^{∞}- Their completeness
- Approximation by smooth functions