# 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

Math 541 is currently the first half of a two-semester sequence on Lebesgue integration in Euclidean space and several related topics, but it is proposed that it become a one-semester course specifically on Lebesgue integration in Euclidean space and Fourier transforms.

### Prerequisites

Currently, Math 541 requires a semester of single-variable real analysis and a semester of multi-variable Calculus. Replacing these prerequisites by Math 316 would imply that the new version of Math 541 could presuppose that students had been exposed to the geometry of **R**^{n} and to metric spaces, which would make it easier to cover the core topics listed below.

### 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^{∞}- Completeness
- Approximation by smooth functions
- Continuity of translation

- Fourier transform on
**R**^{n}- Convolutions
- Basic properties of Fourier transforms
- Composition with translation, dilation, inversion, differentiation, convolution, etc.
- Regularity of transformed functions
- Riemann-Lebesgue Lemma

- Inversion Theorem for L
^{1} - Schwartz class
- Fourier-Plancherel Transform on L
^{2}- Its inversion
- Isomorphism

- Fourier series
- Dirichlet and Fejér kernels
- L
^{2}convergence - Pointwise convergence
- Convergence of Cesàro means