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

From MathWiki

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

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

− | # L<sup>1</sup>, L<sup>2</sup>, and L<sup>∞</sup> | + | # L<sup>1</sup>, L<sup>2</sup>, and L<sup>∞</sup> and normed linear spaces |

− | #* | + | #* Completeness |

#* Approximation by smooth functions | #* Approximation by smooth functions | ||

</div> | </div> |

## Revision as of 09:45, 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^{∞}and normed linear spaces- Completeness
- Approximation by smooth functions