By Artstein-Avidan S.

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If E is locally compact, F X is generated by X − (K ), K ∈ K. Measurability of special multifunctions If X is a random convex weakly compact subset of a Banach space (so that almost all realisations of X are weakly compact convex sets in E), then it is possible to provide a simpler criterion for the measurability of X. 5 (Measurability of convex-valued multifunction). e. the support function of X h(X, u) = sup{ x, u : x ∈ X} is a random variable for each continuous linear functional u ∈ E∗ , where x, u denotes the value of u at x .

K n ∈ K. Then the elements of V are compact in the Fell topology and V ⊂ V. 27). ,Vn ∈ V with V = G 0 ∪ K 0 , G 0 ∈ G and K 0 ∈ K. There exists a sequence {G k , k ≥ 1} of open sets such that G k ↓ K 0 and G k ⊃ cl(G k+1 ) ∈ K for all k ≥ 1. Hence V is a limit of a decreasing sequence of open sets G 0 ∪ G k . Similarly, for each i ∈ {1, . . , n}, Vi can be obtained as a limit of an increasing sequence {K ik , k ≥ 1} of compact sets. Deﬁne G ∪G k . ,K nk Then Yk ∈ V and Yk ↑ Y as k → ∞. In order to show that P(Yk ) ↑ P(Y) note that T (V ∪ Vi ) − P(Y) = −T (V ) + T (V ∪ Vi1 ∪ Vi2 ) + · · · , i1

Fn ∈ F such that X (ω) = Fi for all ω ∈ Ai , 1 ≤ i ≤ n. It is known (see Appendix B) that the space F is separable in the Fell topology if E is LCHS. The separability of F ensures that in this case each random closed set is an almost sure limit (in the Fell topology) of simple random sets. For a general separable metric space E, this is not always the case. 9 (Approximable random sets). A random closed set X is called approximable (with respect to some topology or metric on F ) if X is an almost sure limit of a sequence of simple random closed sets (in the chosen topology or metric).