By A Grothendieck

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S. 2 Measurability and selections 39 Proof. Note that ϕ(ω, x) = (ω, ζx (ω)) is measurable with respect to the product σ -algebra F ⊗ B(E), whence {(ω, x) : ϕ(ω, x) ∈ B} ∈ F ⊗ B(E) for every B ∈ B(E ). The proof is ﬁnished by observing that Graph(Z ) = Graph(X) ∩ ϕ −1 (Graph(Y )) is a measurable subset of Ω × E. 2. 26 for the second one. 2. The minimum α and argmin inside X of a random function ζ . 27 (Measurability of inﬁmum). Let X be an almost surely non-empty random closed set in Polish space E and let ζx be an almost surely continuous stochastic process with values in R.

5(iii), the σ -algebra B(K) is generated by {K ∈ K : K ∩ G = ∅} for G ∈ G. e. the Borel σ -algebra on K coincides with the trace of B(F ) on K. Therefore, these two natural approaches to deﬁne a random compact set produce the same object if E is locally compact. 30 is consistently used to deﬁne random compact sets. s. is understood as sup{P {X ∈ Y} : Y ∈ B(F ), Y ⊂ K} = 1 . The following result is a sort of “tightness” theorem for distributions of random compact sets. 31 (Tightness for random compact sets).

Let Q be the family of all ﬁnite subsets of a countable dense set Q in E. For each F ∈ F it is possible to ﬁnd a countable set F = {x 1 , x 2 , . . } ⊂ F such that F = cl F . Let Fn be a set from Q such that ρH (Fn , Fn ) < n −1 , where Fn = {x 1 . . , x n }. Then Fn converges to F in Wijsman topology, since ρ(x, Fn ) − n −1 ≤ ρ(x, Fn ) ≤ ρ(x, Fn ) + n −1 for each x ∈ E, so that it sufﬁces to notice that ρ(x, Fn ) → ρ(x, F) as n → ∞. 2) {F ∈ F : |ρ(x, F) − ρ(x, F0 )| < r } for x ∈ Q, F0 ∈ F and positive rational r .