Measurable Function


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measurable function

[′mezh·rə·bəl ′fəŋk·shən]
(mathematics)
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.

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(i) For any nonnegative measurable function f(x) in R, we have the following inequality:
For a finite valued function u [member of] [L.sup.0] ([SIGMA]), the weighted composition operator W on [L.sup.p] ([SIGMA]) with 1 [less than or equal to] p [less than or equal to] [infinity], induced by u and the non-singular measurable function [phi] is given by W = [M.sub.u] [??] [C.sub.[phi]] where [M.sub.u] is a multiplication operator and [C.sub.[phi]] is a composition operator on [L.sup.p] ([SIGMA]) defined by [M.sub.u]f = uf and [C.sub.[phi]]f = f [??] [phi], respectively.
where a and b are Lebesgue measurable essentially bounded on [0, [infinity]) functions, and there exists [tau] > 0 such that the measurable function h satisfies 0 [less than or equal to] t - h(t) [less than or equal to] [tau].
If [[iota].sup.2] is measurable function, then A = [([[iota].sup.2]).sup.-1] ([N.sub.e]) is a measurable set, where [N.sub.e] = {x [member of] B(H) : [absolute value of (<xe, e>)] < 1}.
Where [mathematical expression not reproducible] and f : J x E x E x [OMEGA] [right arrow] E are given continuous functions, ([OMEGA], A, v) is a measurable space, and E is a real (or complex) Banach space with norm [[parallel] * [parallel].sub.e] and dual [E.sup.*], such that E is the dual of a weakly compactly generated Banach space [mathematical expression not reproducible], is the left-sided mixed Hadamard integral of order r, and [mathematical expression not reproducible] is a given continuous and measurable function such that
Let f : X [right arrow] [0, [infinity]) be a measurable function (i.e., {f [greater than or equal to] [alpha]} [member of] A for any [alpha] [greater than or equal to] 0).
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