#### Prerequisites & Bulletin Description

#### Course Objectives

The aim of Analysis I is to introduce students to the theory and applications of Lebesgue integration and the theory of differentiation of absolutely continuous functions and functions of bounded variation.

Students are expected to master the proofs of the principal results of Lebesgue measure and integration. They are also expected to apply the ideas, techniques and results of this theory to specific examples.

Students who successfully complete this course should be capable of understanding and applying the concepts of outer measure, measurable sets, measurable functions, integration, modes of convergence, bounded variation, absolute continuity, and differentiation. They will devise, organize, and present their solutions in correct mathematical English.

#### Evaluation of Students

Students will be graded on their ability to devise, organize and present in correct mathematical English rigorous solutions to assignments and problems on exams. While instructors may design their own methods of evaluating student performance, these methods must include in-class examinations, assignments and a final examination.

#### Course Outline

Topics |
Number of Weeks |

Lebesgue measure | 4 |

The integral | 4 |

Convergence theorems | 3 |

Differentiation | 3 |

Applications and generalizations | Include as time permits |

#### Textbooks & Software

*Real Analysis (3rd Edition) *, by Royden

*Measure and Integration *, by Wheeden and Zygmund

Submitted by: Eric Hayashi

Date: September 18, 2008