On Deterministic and Random Fixed Points

Abstract

Based on an extension of Aumann’s measurable selection theorem due to Leese, it is shown that each fixed point theorem for $F(\omega , \cdot )$ produces a random fixed point theorem for $F$ provided the $\sigma$-algebra $\Sigma$ for $\Omega$ is a Suslin family and $F$ has a measurable graph (in particular, when $F$ is random continuous with closed values and $X$ is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended.