# Math 541: Real Analysis

From MathWiki

## 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