For any given measurable mapping, the multivalued mappings are measurable by Lemma 2.5.
Let E be a nonempty set, S be a measurable space which consists of some metrics on E, ((Omega, P)) be a complete probabilistic measure space and (f : Omegarightarrow S) be a measurable mapping.
Let X be a nonempty set, S be a measurable space which consists of some metrics on the X, ((Omega, P)) be a complete probabilistic measure space and (f : Omegarightarrow S) be a measurable mapping.
It is not hard to prove that any bounded measurable mapping which is ⪯-comonotonic with all the mappings of ℛ belongs to it, and so we conclude that the usual multivariate stochastic order is generated by the componentwise order.
Let E be a linear space, S be a measurable space which consists of some norms on E, ((Omega, P)) be a complete probabilistic measure space and (f : Omegarightarrow S) be a measurable mapping.
Assume that α : Ω → (0, 1] is a measurable step size function. For any measurable mapping x0 : Ω → X, the set-valued mappings S ., x0, T ., x0, P ., x0, Q(., x0, G(., x0: Ω → CB X) are measurable by Lemma 3.4.
Example Let X be a nonempty set, S be a measurable space which consist of some metrics on the X, ( Ω, P ) be a complete probabilistic measure space and f : Ω → S be a measurable mapping.
The so-called birandom variable is a measurable mapping from a probability space to a collection of random variables.
Fuzzy random variable is a measurable mapping from a probability space to a collection of fuzzy variables.
That function is a measurable mapping b : X → [ 1, ∞ ).
A measurable mapping is also called a random variable.
