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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Extreme limits of compacta valued functions


Author: T. F. Bridgland
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1972-0362209-2
Keywords: Set valued functions, trajectory integral, control theory, multivalued functions, compacta valued functions, generalized differential equations
Article copyright: © Copyright 1972 American Mathematical Society