Measurable Function

(in the original meaning), a function f(x) that has the property that for any t the set E_t of points x, for which each f(x) ≤ t, is Lebesgue measurable. This definition of a measurable function was given by the French mathematician H. Lebesgue. The sum, difference, product, and quotient of two measurable functions, as well as the limit of a sequence of measurable functions, are in turn measurable functions. Thus, the basic operations of algebra and analysis do not go beyond the framework of the set of measurable functions. Russian and Soviet mathematicians have made a major contribution to the study of measurable functions (D. F. Egorov, N.N. Luzin, and their students). Luzin proved that a function is measurable if and only if it can be made continuous after its values are varied in a set of as small as desired measure. This is the so-called C-property of measurable functions.

In the abstract theory of measure, the function f(x) is said to be a measurable function with respect to some measure μ. if the set E_t is found in the domain of definition of the measure μ. In modern probability theory measurable functions are called random variables.