Measurable Function

(redirected from Measurable mapping)

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.

References in periodicals archive ?
Let [x.sub.0], [y.sub.0], [z.sub.0],[u.sub.0] : [OMEGA] [right arrow] C be an arbitrary measurable mapping for to [omega] [member of] [OMEGA], for n = 1, 2, ....
Let [x.sub.0], [y.sub.0], [z.sub.0], [u.sub.0] : [OMEGA] [right arrow] C be an arbitrary measurable mapping for [omega] [member of] [OMEGA], n = 0, 1, 2, ....
We say that [??] has a random fixed point [xi] if there exists a measurable mapping [xi]: [OMEGA] [right arrow] A such that:
Let x: [OMEGA] [right arrow] X be a measurable mapping; then the mapping [xi]: [OMEGA] [right arrow] X defined by [xi]([omega]) = T([omega],x([omega])) is measurable.
where h : [??]0, [infinity]) [right arrow] [??]0, [infinity]) is a Lebesgue measurable mapping which is summable (i.e., with finite integral) on each compact subset of [0, [infinity]), such that, for every [epsilon] > 0 we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] h(t)dt > 0.
where [mathematical expression not reproducible], and [x.sub.0] : [OMEGA] [right arrow] [F.sup.b.sub.c]([R.sup.d]) are appropriately measurable mappings while W is the standard real Wiener process.
Let a : [0, T] x [??] [right arrow] [R.sup.n] and [alpha]: [0,T] x [??][right arrow] [[??].sub.+](n) be measurable mappings. The first problem is to find a stochastic process [xi](t) whose forward and quadratic P-mean derivatives at each t are a(t, [xi](.)) and (t, [xi](.)), respectively.