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.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

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References in periodicals archive

"I believe this is an important milestone in the field of mRNA therapeutics as it starts to address many questions regarding the safety and delivery of mRNA to human tissues, the duration and level of the protein that can be expressed and the ability of the technology to have a physiologic, measurable function over a prolonged period of time," said Kenneth Chien, M.D., Ph.D., a professor in the Department of Cell and Molecular Biology and the Integrated Cardio Metabolic Center at the Karolinska Institute in Stockholm, a Moderna scientific co-founder and co-author on the paper.

Moderna announces publication of Phase 1a/b study in Nature Communications

The exponent p(x) is a given measurable function on [bar.[OMEGA]] such that

Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities

Variable Lebesgue spaces are a generalization of the Lebesgue spaces that allow the exponents to be a measurable function and thus the exponent may vary.

Compact Operators on the Bergman Spaces with Variable Exponents on the Unit Disc of C

A measurable function f : [OMEGA] [right arrow C is called a random fixed point for the operator T : [OMEGA] x C [right arrow] C if T([omega], f([omega])) = f([omega]).

On the Equivalence of Stochastic Fixed Point Iterations for Generalized [phi]-Contractive-Like Operators

(i) For any nonnegative measurable function f(x) in R, we have the following inequality:

Equivalent Property of a Hilbert-Type Integral Inequality Related to the Beta Function in the Whole Plane

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.

Stability constants for weighted composition operators on [L.sup.p]([SIGMA])

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].

STABILITY OF TIME-DEPENDENT DYNAMIC MONOPOLY WITH CONCENTRATED AND DISTRIBUTED DELAYS

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}.

Borel structures coming from various topologies on B(H)

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

Weak Solutions for Partial Random Hadamard Fractional Integral Equations with Multiple Delays

(2) For any bounded, Borel measurable function g(x),

Sample dependence in the maximum entropy solution to the generalized moment problem

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).

Optimization in a Fuzzy Environment

Let {P(x,y,A) : x, y [member of] X, A [member of] F} be a family of functions on X x X x F such that, for any fixed x,y [member of] X P(x, y,*) [member of] S(X, F), P(x, y, A) regarded as a function of two variables x and y with fixed A [member of] F is a measurable function on (X x X, F [cross product] F) and P(x, y, A) = P(y, x, A) for any x, y [member of] X and A [member of] F.

On poisson nonlinear transformations