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.
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 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.
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.
Let (x:Omegato H) be a measurable mapping and (theta:Omegato 0,+infty)) be a measurable function, then for all (x t)in H) and for each (tinOmega), (R_{theta(t)}^{phi} t,x t))) is a gap function for the RGVIP (2.1).
Let g : X → R be a measurable mapping satisfying that g and f are ⪯-comonotonic for all f ∈ R. Let us see that given x, y ∈ X with x ⪯ y ( x ≠ y ) there exists a mapping l ∈ R with l ( x ) < l ( y ), which implies that g ( x ) ≤ g ( y ), and so g is ⪯-preserving, that is, g ∈ R. If the above result is false, there exist x, y ∈ X with x ≠ y such that x ⪯ y and l ( x ) = l ( y ) for all l ∈ R.
Lemma 3.8 (Dzedzej and Gelman [36]) Let F: [0, α] → Pc, cp(ℝ n ) be a measurable map such that the Lebesgue measure μ of the set {t: dim F t) < 1} is zero.
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, ∞ ).
