Extension of random contractions

Abstract:

Let $\Omega$ be a measurable space. Let $X$ and $Y$ be separable Hilbert spaces and let $D$ be a subset of $X$. Then every random contraction $f:\Omega \times D \to Y$ can be extended to a random contraction defined on all $\Omega \times X$. This statement remains true if $\Omega$ is a complete measurable space, $X$ and $Y$ are separable metric spaces and the pair $(X,Y)$ has the Kirszbraun intersection property.