EXTENSION-THEOREMS OF CONTINUOUS RANDOM LINEAR-OPERATORS ON RANDOM DOMAINS

The central purpose of this paper is to prove the following theorem: let (Omega, sigma, u) be a complete probability space, (B, parallel to .parallel to) a normed linear space over the scalar field K, E: Omega --> 2(B) a separable random domain with linear subspace values, and f: GrE --> K a continuous random linear operator, where GrE = {(omega, x) is an element of Omega X B\x is an element of E(omega)} denotes the graph of E. Then there exists a continuous random linear operator (f) over tilde: Omega X B --> K such that (f) over tilde(omega, x) = f(omega, x) For All omega is an element of Omega, x is an element of E(omega), and sup{\(f) over tilde(omega, x)\ \x is an element of B, parallel to x parallel to less than or equal to 1} = sup{\f(omega, x)\ \x is an element of E(omega), parallel to x parallel to less than or equal to 1}, for each omega in Omega. For the case where E is not separable, a result similar to the above-stated theorem is also given, which generalizes and improves many previous results on random generalizations of the Hahn-Banach Theorem. (C) 1995 Academic Press, Inc.