Remarks on the extension of random contractions

Abstract

for x1; x2 ∈D, where D is a nonempty subset of X . The problem of the existence of an extension of any contraction mapping is an interesting one. This problem was studied for Euclidean spaces by Kirzbraun [7] and extended to Hilbert spaces by Valentine [15, 16]. Recently, Myjak and Zygadlewicz [11], Beg and Shahzad [3], and Shahzad [14] gave stochastic versions of the Kirzbraun–Valentine extension theorem. Let ( ; ) be a measurable space with a -algebra of subsets of . Denote by I and F the sets of in nite and nite sequences of positive integers, respectively. Let G be a family of sets and k :F → G be a map. For each r = (ri)i=1 ∈I and n∈N, we shall denote r|n by (r1; : : : ; rn), then ⋃ r ∈I ⋂∞ n=1 k(r|n) is said to be obtained from G by the Souslin family. If every set obtained from G in this way is also in G, then G is called a Souslin family. If is an outer measure on ( ; ) the is a Souslin family. In particular, if ( ; ) is a complete measurable space, then is a Souslin family (see [17]). Let 2X be the family of all subsets of X and C(X ) the family of all nonempty closed subsets of X . A multifunction F : → 2X \{ } is said to be measurable (resp. weakly measurable) if for every closed (resp. open) subset A of X , the set F−1(A) = {!∈ : F(!)∩A 6= }∈ . Note that if X is a metric space measurability implies weak