  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 -dimensional euclidean space . for instance, the lebesgue measure of the interval  in the real numbers is its length in the everyday sense of the word&thinsp;â&thinsp;specifically, 1. technically, a measure is a function that assigns a non-negative real number or +â to (certain) subsets of a set  (see definition below). it must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) 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. this problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a -algebra. this means that countable unions, countable intersections and complements of measurable subsets are measurable. non-measurable sets in a euclidean space, on which the lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.halmos, paul (1950), measure theory, van nostrand and co. indeed, their existence is a non-trivial consequence of the axiom of choice. measure theory was developed in successive stages during the late 19th and early 20th centuries by Ãmile borel, henri lebesgue, johann radon and maurice frÃ©chet, among others. the main applications of measures are in the foundations of the lebesgue integral, in andrey kolmogorov's axiomatisation of probability theory and in ergodic theory. in integration theory, speci
Measure_theory Measures_(measure_theory)