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.
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- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 170 (1972), 149-163
- MSC: Primary 54C60; Secondary 28A45
- DOI: https://doi.org/10.1090/S0002-9947-1972-0362209-2
- MathSciNet review: 0362209