probability space


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probability space

[‚präb·ə′bil·əd·ē ‚spās]
(mathematics)
A measure space such that the measure of the entire space equals 1.
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Probability theory is a branch of mathematics that allows to evaluate the possibility of occurrence of an event whose probable results can be analyzed in a probability space [14,15].
Given a probability space ([OMEGA], A, P) and a hyper-measurable structure (P(X), B) on X, a neutrosophic random set on X (see [8]) is defined to be a triple [xi] := ([[xi].sub.T], [[xi].sub.I], [[xi].sub.F]) in which [[xi].sub.T], [[xi].sub.I] and [[xi].sub.F] are mappings from [OMEGA] to P(X) which are A-B measurables, that is,
We always assume that ([OMEGA], F, P) is a completed probability space, W =: {[W.sub.t] : t [greater than or equal to] 0} is a real-valued Brownian motion defined on ([OMEGA], F, P), and {[F.sub.t] : t [greater than or equal to] 0} is natural filtration generated by the Brownian motion W; i.e., for any t [greater than or equal to] 0
Convergence of an approximation to a strong solution on a given probability space was established by Gyongy and Krylov in [16] using coupling.
(ii) Let ([OMEGA], A, P) be a probability space; that is, [OMEGA] alone is called the sample space, A is a [sigma]-algebra on [OMEGA], and P is a probability measure on ([OMEGA], A).
Till the end of the section let ([OMEGA], S, P) be a probability space (i.e., [OMEGA] is a nonempty set, S is a [sigma]-algebra of subsets of [OMEGA], and P : S [right arrow] [0,1] is a probability measure).
where [a.sub.i](t) = [a.sub.i](t; w) :]0, +[infinity][[x[OMEGA] [right arrow] R, 1 [less than or equal to] i [less than or equal to] 3 and f (x) = f (x; w) : R x [OMEGA] [right arrow] R are s.p.'s, defined in a complete probability space ([OMEGA], F, P), that satisfy certain hypotheses that will be specified later.
Let ([mathematical expression not reproducible]) be a complete probability space. Here, [mathematical expression not reproducible] is a cadlag, [mathematical expression not reproducible] is the time limit for uncertain economic system, and Q is the equivalent martingale measure.
We start by building a probability space on the initiative knowledge, and we will see how beliefs vary.
The fundamental ingredients of probability theory are the probability space ([OMEGA], [summation], [mu]) and the abelian von Neumann algebra [L.sup.[infinity]]([OMEGA], [mu]), where [OMEGA] is a set, [summation] is [sigma]-algebra of measurable subsets of [OMEGA], [mu] is a probability measure, and [L.sup.[infinity]]([OMEGA], [mu]) is the set of all essentially bounded measurable functions.
The Birkhoff-Kinchin Ergodic Theorem (See Billingsley [10]) states that if [theta] is a measure preserving transformation defined on a probability space ([OMEGA], F, P) and X : ([OMEGA], F) [right arrow] ([R.sup.d], B([R.sup.d])) is an integrable random vector, then

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