Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables

Abstract

Abstract. A theory of random Borel sets is presented, based on dyadic resolutions of compact metric spaces. The conditional expectation of the intersection of two independent random Borel sets is investigated. An example based on an embedding of Sierpiński's universal curve into the space of Borel sets is given. 2000

Integration in a dynamical stochastic geometric framework

“…ω ∈ Ω. From the first part of proof and since X is reflexive (A-0), we get that G a,b is closed (see ( [25], Thm. 2.2.3)).…”

“…A RaCS X is an element of U[Ω, F, P; F ]. It can be proved (see [25]) that, if X, X 1 , X 2 are RaCS and if ξ is a measurable real-valued function, then X 1 ⊕ X 2 , X 1 X 2 , ξX and (Int X) C are RaCS. Moreover, if {X n } n∈N is a sequence of RaCS then X = n∈N X n is a RaCS, too.…”

“…One may define Ω Xdμ as a limit of integrals of simple RaCS, see [25], Def. 2.2.4 (Bochner integral)).…”

Existence of solutions for fractional interval--valued differential equations by the method of upper and lower solutions

“…kf .t /k belongs to L P .J I R/. In fact, we readily get from [17] that L p .J I R.I // is a complete metric space with respect to the metric H p defined by H p .f; g/ WD kf « gH gk p for any f; g 2 L p .J I R.I //, where…”

“…R C WD OE0; 1/ will be defined by Indeed, it is obviously that the Hausdorff metric H is associated with the norm k k by kAk D H .A; f0g/. Moreover, it follows from Li, Ogura and Kreinovich in [17], which knows that .R.I /; H / is a complete, locally compact and separable metric space. For any A 2 R.I /, it can be seen A C .…”