Measurable Function

Also found in: Wikipedia.

measurable function

[′mezh·rə·bəl ′fəŋk·shən]
A real valued function ƒ defined on a measurable space X, where for every real number a all those points x in X for which ƒ(x) ≥ a form a measurable set.
A function on a measurable space to a measurable space such that the inverse image of a measurable set is a measurable set.

Measurable Function


(in the original meaning), a function f(x) that has the property that for any t the set Et 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 Et is found in the domain of definition of the measure μ. In modern probability theory measurable functions are called random variables.

References in periodicals archive ?
3], recall that a measurable function f : [OMEGA] [right arrow] C belongs to [L.
Y]), mapping two measurable neutrosophic spaces, is called neutrosophic measurable function if [for all]B [member of][[SIGMA].
Every essentially bounded complex valued measurable function [f.
A question posed by Lusin in 1915 asks whether it be possible to find for every measurable function [0,2[pi]] a trigonometric series, with coefficient sequence converging to For real-valued functions, this question was given an affirmative answer by Men'shov [8] in 1941.
Writing Measurable Function and Transition IEP Goals
Let f : X [right arrow] R be a measurable function.
there exists at least one measurable function h: [0, T] [right arrow] X such that h(t) [member of] H (t) a.
d] [right arrow] R a compactly supported bounded measurable function and B, W and P, Q four non-empty compact subsets of[R.
If f is a non-negative and measurable function with the Laplace transform F, then