MEASURE

in mathematics, generalization of the concepts of length and area to arbitrary sets of points not composed of intervals or rectangles. Abstractly, a measure is any rule for associating with a set a number having the properties of being non-negative and additive; i.e., the measure of the union of two non-overlapping sets is equal to the sum of their individual measures. The measure of an elementary set composed of a finite number of rectangles can be defined simply as the sum of their areas found in the usual manner. For other sets, such as curved regions or vaporous regions with missing points, the concepts of outer and inner measure must first be defined. The outer measure of a set is the number that is the lower bound of the area of all elementary rectangular sets containing the given set, while the inner measure of a set is the upper bound of the areas of all such sets contained in the region. If the inner and outer measures of a set are equal, this number is called its measure, and the set is said to be measurable. The measure of a set of points on a line is defined similarly using intervals in place of rectangles. For example, the set of rational numbers from 0 to 1 is not composed of a finite number of intervals, and so no length is defined for it. It has a measure, however, that can be found in the following way: The rational numbers are countable, and each successive number can be covered by intervals of length 1/8, 1/16, 1/32, . . . etc., the total sum of which is 1/4, calculated as the sum of the infinite geometric series. The rational numbers could also be covered by intervals of lengths 1/16, 1/32, 1/64, . . . etc., the total sum of which is 1/8. By starting with smaller and smaller intervals, the total length of intervals covering the rationals can be reduced to smaller and smaller values approaching the lower bound of zero, and so the outer measure is zero. The inner measure is always less than or equal to the outer measure, so it must also be zero. Therefore, the rationals are measurable with measure zero.