Multi-valued mapping
point-to-set mapping
A mapping associating with each element
of a set
a subset
of a set
. If for each
the set
consists of one element, then the mapping
is called single-valued. A multi-valued mapping
can be treated as a single-valued mapping of
into
, that is, into the set of all subsets of
.
For two multi-valued mappings ,
, their inclusion is naturally defined:
if
for all
. For any family of multi-valued mappings
,
, the union and intersection are defined:
if
for all
and
if
for all
. For any family of multi-valued mappings
,
, the multi-valued mapping
is called the Cartesian product of the multi-valued mappings
if
. A section of a multi-valued mapping
is a single-valued mapping
such that
for all
. The graph of a multi-valued mapping
is the set
.
A multi-valued mapping of a topological space
into a topological space
is called upper semi-continuous if for every open set
the set
is open in
, or equivalently: For any
and any neighbourhood
of
there is a neighbourhood
of
such that
, where
. A multi-valued mapping from a topological space
to a topological space
is called lower semi-continuous if for any open set
the set
is open in
. If a multi-valued mapping satisfies both properties simultaneously, then it is called a continuous multi-valued mapping.
Let be a topological vector space. A multi-valued mapping
is called convex-compact valued if
is a convex compact set for all
. For a finite set of multi-valued mappings
,
, the algebraic sum
is defined by
. The intersection of any (finite) family of upper semi-continuous (respectively, continuous) multi-valued mappings is upper semi-continuous (respectively, continuous). The Cartesian product of a finite family of upper semi-continuous multi-valued mappings is upper semi-continuous. The algebraic sum of a finite family of upper semi-continuous (convex-compact valued) mappings is upper semi-continuous (convex-compact valued). The intersection and Cartesian product of any family of convex-compact valued mappings is convex-compact valued.
Let be a paracompact space and
a locally convex metric linear space (cf. Locally convex space; Linear space; Metric space). Let
be a multi-valued mapping which is upper semi-continuous and is such that
is closed in
for every
. Then the multi-valued mapping
admits continuous sections. Let
and
be spaces with given
-algebras
and
; a multi-valued mapping
is called measurable if the graph
belongs to the smallest
-algebra
of
containing all sets of the form
, where
and
. If
is a measurable multi-valued mapping from
to a complete separable metric space
, where
is the Borel
-algebra of
, then
has measurable sections
.
References
[1] | K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French) |
Comments
A multi-valued mapping is also called a set-valued or many-valued mapping. Sections are also called selections.
Theorems which prove that certain kinds of multi-valued mappings admit selections are called selection theorems. The measurable selection theorem stated in the last sentence of the main article above is known as von Neumann's measurable choice theorem. A number of selection theorems and some applications are discussed in [a4].
References
[a1] | E. Michael, "Continuous selections" Ann. of Math. , 63 (1956) pp. 361–382 |
[a2] | E.A. Michael, "A survey of continuous selections" W.M. Fleischmann (ed.) , Set valued mappings, selections and topological properties of ![]() |
[a3] | K. Przeslawski, D. Yost, "Continuity properties of selectors and Michael's Theorem" Mich. Math. J. , 36 (1989) pp. 113–134 |
[a4] | T. Parthasarathy, "Selection theorems and their applications" , Lect. notes in math. , 263 , Springer (1972) |
Multi-valued mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-valued_mapping&oldid=15886