# Measure and Integration: Course Contents (MAT 452)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

## Course Objective

At the end of this course the students will be able to understand basic notions of measure and integration theory. Theorems regarding algebra of measurable functions and integrals will also discussed.

## Course Contents (MAT 452)

• Definition and examples of algebras and s-algebras,
• Basic properties of measurable spaces, Definition and examples of measure spaces, Outer measure,
• Lebesgue measure, Measurable sets, Complete measure spaces, Some equivalent formulations of measurable functions, Examples of measurable functions, Various characterization of measurable functions,
• Property that holds almost everywhere, Egorov’s theorem,
• Definition of Lebesgue integral, Basic properties of Lebesgue integrals,
• Comparison between Riemann integration and Lebesgue integration, L2-space,
• The Riesz-Fischer theorem.