Definition of the metric on the space $\mathrm{clos}_{\varnothing}(X)$
of closed subsets of a metric space $X$
and properties of mappings with values in $\mathrm{clos}_{\varnothing}(\mathbb{R}}^n)$
Abstract:
The paper is concerned with the extension of tests for superpositional measurability,
Filippov's implicit function lemma and the Scorza Dragoni property to set-valued
(and, as a corollary, to single-valued) mappings that fail to satisfy
the Carathéodory conditions (the upper Carathéodory conditions) and are not continuous (upper semicontinuous) in the phase variable. To obtain the corresponding results the
space $\mathrm{clos}_{\varnothing}(X)$ of all closed
subsets (including the empty set) of an arbitrary metric space $X$
is introduced;
a metric on $\mathrm{clos}_{\varnothing}(X)$ is proposed; the space $\mathrm{clos}_{\varnothing}(X)$ is shown to be complete whenever the
original space $X$ is; a criterion for convergence of a sequence is put forward; mappings with values in
$\mathrm{clos}_\varnothing(X)$ are studied. Some results on
set-valued mappings satisfying the Carathéodory conditions
and having compact values in $\mathbb R^n$ are shown to hold for
mappings with values in $\mathrm{clos}_\varnothing(\mathbb R^n)$, measurable in the first argument, and continuous in the proposed metric in the second argument.
Bibliography: 22 titles.
Keywords:superpositional measurability, Filippov's implicit function lemma,
Scorza Dragoni property, the space of closed subsets of a metric space,
set-valued mapping.
Fulltext:
PDF file (688 kB)
