Invariant Measures

Abstract

Abstract A measure is a generalization of what we mean by area in the plane and volume in three dimensions. A measure assigns a “size” to a set. Thus, for example, the measure of disjoint sets should be the sum of the measures of each. It turns out that there are some sets so “bad” that they cannot be assigned a measure in a natural way. The collection of “‘good” sets forms what is called a sigma algebra and a measure assigns a nonnegative number to each set in the sigma algebra. Measure and integration theory are standard topics taught in first-year graduate mathematics courses in analysis. Some definitions that are needed in the next section are given below for reference. The reader who is not familiar with measure theory may just skip this section and keep in mind that good sets in a Euclidian space are approximated by disjoint unions of generalized boxes, and that the volume of a box is the product of the lengths of its edges.