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On simultaneous extension of continuous partial functions

Authors:	Hans-Peter A Kü nzi Leonid B Shapiro

Institution:	Department of Mathematics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland ; Department of Mathematics, Academy of Labor and Social Relations, Lobachevskogo 90, 117454 Moscow, Russia

Abstract:	For a metric space let ${\cal C}_{vc}(X)$ (that is, the set of all graphs of real-valued continuous functions with a compact domain in ) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of $X\times {\mathbf {R}}.$ It is shown that there exists a continuous mapping $\Phi :{\cal C}_{vc}(X)\rightarrow {\cal C}_b(X)$ satisfying the following conditions: (i) $\Phi (f)\vert \operatorname {dom}f= f$ for all partial functions (ii) For every nonempty compact subset of $\Phi \vert{\cal C}_b(K):{\cal C}_b(K) \rightarrow {\cal C}_b(X)$ is a linear positive operator such that $\Phi (1_K)=1_X$ .

Keywords:	Extension of function partial function compact domain Hausdorff metric Lipschitzian function probability measure

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