Measurable Set

(in the original meaning), a set to which the definition of measure given by the French mathematician H. Lebesgue is applicable. Measurable sets are one of the principal concepts of the theory of functions of a real variable and are the most important and an extremely broad class of point sets. In particular, closed sets and open sets lying on some segment are measurable sets. In the abstract theory of measure, sets belonging to the domain of definition μ are said to be measurable with respect to some measure μ. In the case when μ is a probability distribution, measurable sets are also called random events.