Mathematics: Thry Functions Real Variables

Lebesgue measure and integration theory, Monotone and Dominated Convergence Theorems and applications, product measures, basic theory of LP-spaces. Prerequisite: Two semesters of real analysis required.

Prereqs: MATH 3472; Open to Degree and PACE students

The course will introduce the student to the theory of measure-based integration, with special emphasis on Lebesgue measure and the Lebesgue integral, which generalizes the Riemann integral. The class text will be The Elements of Integration and Lebesgue Measure by Robert Bartle. For a different point of view, there is also a supplemental text Measure, Integration and Real Analysis by Sheldon Axler. The electronic version of the Axler book is available for FREE at https://link.springer.com/content/pdf/10.1007%2F978-3-030-33143-6.pdf After some motivation, showing the inadequacy of the Riemann integral, we will develop the abstract theory of measure-based integration, including the basic theory of L^p spaces and the major convergence theorems. Next we will show how measures and their natural domains ("sigma-algebras") are constructed, and show how these apply to the construction of Lebesgue measure (the generalization of length) and the Lebesgue measurable sets on the real line. This work will cover (roughly) chapters 1-7 and 9 of Bartle. If time permits we will treat measure decompositions, L^p-L^q duality, basic functional analysis, and product measures.

Come to class. Do the reading. Do the homework.

The grade will be based entirely on homework: 7 large homework sets. The sets won't look big, but you will find them time-consuming. There will be no quizzes, midterm, or final exam.