Measure (mathematics)
In mathematical analysis, a
measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the
n-dimensional Euclidean space
Rn. For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to subsets of a set
X. It must assign 0 to the empty set and be additive: the measure of a 'large' subset that can be decomposed into a finite number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a
consistent size to
each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.