Extreme limits of compacta valued functions

Abstract:

Let $X$ denote a topological space and $\Omega (X)$ the space of all nonvoid closed subsets of $X$. Recent developments in analysis, especially in control theory, have rested upon the properties of the space $\Omega (X)$ where $X$ is assumed to be metric but not necessarily compact and with $\Omega (X)$ topologized by the Hausdorff metric. For a continuation of these developments, it is essential that definitions of extreme limits of sequences in $\Omega (X)$ be formulated in such a way that the induced limit is topologized by the Hausdorff metric. It is the purpose of this paper to present the formulation of such a definition and to examine some of the ramifications thereof. In particular, we give several theorems which embody “estimates of Fatou” for integrals of set valued functions.