measure theory

measure theory

In mathematics, a generalization of the concepts of length and area (see length, area, and volume) to arbitrary sets of points not composed of line segments or rectangles. A measure is any rule for associating a number with a set. The result must be nonnegative and also additive, meaning that the measure of two nonoverlapping sets equals the sum of their individual measures. This is simple enough for sets consisting of line segments or rectangles, but the measure of sets such as curved regions or intervals with missing points requires more abstract methods, including limits and upper and lower bounds.