MATH242

Description

How do you measure the length of an arbitrary subset of numbers? For intervals, the answer is easy; but what about for the set of rational numbers, which are "everywhere and nowhere" on the real number line? The branch of mathematics known as measure theory is a deep investigation of this question. After introducing the basic properties of a "measure function," this course will concentrate on a particular type, known as "Lebesgue measure," which plays a central role in many areas of mathematics. To begin with, it allows for a new and better theory of integration, superseding the "Riemann integral" that you learned in calculus. It also allows one to give a rigorous grounding of probability theory, randomness, the law of large numbers, etc. Time permitting, additional topics may include, among other things, the use of measure theory in the field of ergodic theory, which has its origins in statistical physics. Therefore, this course will be of interest not only to mathematics majors, but also to students in physics, computer science, and economics. Formally MATH 214