View source for Multi-valued mapping

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''point-to-set mapping''

A mapping  $  \Gamma :  X \rightarrow Y $
associating with each element  $  x $
of a set  $  X $
a subset  $  \Gamma ( x) $
of a set  $  Y $. 
If for each  $  x \in X $
the set  $  \Gamma ( x) $
consists of one element, then the mapping  $  \Gamma $
is called single-valued. A multi-valued mapping  $  \Gamma $
can be treated as a single-valued mapping of  $  X $
into  $  2  ^ {Y} $, 
that is, into the set of all subsets of  $  Y $.

For two multi-valued mappings  $  \Gamma _ {i} :  X \rightarrow Y $, 
$  i = 1 , 2 $, 
their inclusion is naturally defined:  $  \Gamma _ {1} \subset  \Gamma _ {2} $
if  $  \Gamma _ {1} ( x) \subset  \Gamma _ {2} ( x) $
for all  $  x \in X $. 
For any family of multi-valued mappings  $  \Gamma _  \alpha  :  X \rightarrow Y $, 
$  \alpha \in A $, 
the union and intersection are defined:  $  \Gamma = \cup _ {\alpha \in A }  \Gamma _  \alpha  $
if  $  \Gamma ( x) = \cup _ {\alpha \in A }  \Gamma _  \alpha  ( x) $
for all  $  x \in X $
and  $  \Gamma = \cap _ {\alpha \in A }  \Gamma _  \alpha  $
if  $  \Gamma ( x) = \cap _ {\alpha \in A }  \Gamma _  \alpha  ( x) $
for all  $  x \in X $. 
For any family of multi-valued mappings  $  \Gamma _  \alpha  :  X \rightarrow Y _  \alpha  $, 
$  \alpha \in A $, 
the multi-valued mapping  $  \Gamma = \prod _ {\alpha \in A }  \Gamma _  \alpha  :  X \rightarrow \prod _ {\alpha \in A }  Y _  \alpha  $
is called the Cartesian product of the multi-valued mappings  $  \Gamma _  \alpha  $
if  $  \Gamma ( x) = \prod _ {\alpha \in A }  \Gamma _  \alpha  ( x) $. 
A section of a multi-valued mapping  $  \Gamma $
is a single-valued mapping  $  f :  X \rightarrow Y $
such that  $  f ( x) \in \Gamma ( x) $
for all  $  x \in X $. 
The graph of a multi-valued mapping  $  \Gamma $
is the set  $  G ( \Gamma ) = \{ {( x , y ) \in X \times Y } : {y \in \Gamma ( x) } \} $.

A multi-valued mapping  $  \Gamma $
of a topological space  $  X $
into a topological space  $  Y $
is called upper semi-continuous if for every open set  $  U \subset  Y $
the set  $  \Gamma  ^ {+} ( u) = \{ {x \in X } : {\Gamma ( x) \subset  U,  \Gamma ( x) \neq \emptyset } \} $
is open in  $  X $, 
or equivalently: For any  $  x \in X $
and any neighbourhood  $  U $
of  $  \Gamma ( x) $
there is a neighbourhood  $  O x $
of  $  x $
such that  $  \Gamma ( Ox) \subset  U $, 
where  $  \Gamma ( Ox ) = \cup \{ {\Gamma ( y) } : {y \in Ox } \} $. 
A multi-valued mapping from a topological space  $  X $
to a topological space  $  Y $
is called lower semi-continuous if for any open set  $  U \subset  Y $
the set  $  \Gamma  ^ {-} ( U) = \{ {x \in X } : {\Gamma ( x) \cap U \neq \emptyset } \} $
is open in  $  X $. 
If a multi-valued mapping satisfies both properties simultaneously, then it is called a continuous multi-valued mapping.

Let  $  Y $
be a topological vector space. A multi-valued mapping  $  \Gamma :  X\rightarrow Y $
is called convex-compact valued if  $  \Gamma ( x) $
is a convex compact set for all  $  x \in X $. 
For a finite set of multi-valued mappings  $  \Gamma _ {i} :  X \rightarrow Y $, 
$  i \in I $, 
the algebraic sum  $  \Gamma = \sum _ {i \in I }  \Gamma _ {i} $
is defined by  $  \Gamma ( x) = \sum _ {i \in I }  \Gamma _ {i} ( x) $. 
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  $  X $
be a [[Paracompact space|paracompact space]] and  $  Y $
a locally convex metric linear space (cf. [[Locally convex space|Locally convex space]]; [[Linear space|Linear space]]; [[Metric space|Metric space]]). Let  $  \Gamma :  X \rightarrow Y $
be a multi-valued mapping which is upper semi-continuous and is such that  $  \Gamma ( x) $
is closed in  $  Y $
for every  $  x \in X $. 
Then the multi-valued mapping  $  \Gamma $
admits continuous sections. Let  $  ( X , \mathfrak A ) $
and  $  ( Y , \mathfrak B ) $
be spaces with given  $  \sigma $-
algebras  $  \mathfrak A $
and  $  \mathfrak B $; 
a multi-valued mapping  $  \Gamma :  ( X , \mathfrak A ) \rightarrow ( Y , \mathfrak B ) $
is called measurable if the graph  $  G ( \Gamma ) $
belongs to the smallest  $  \sigma $-
algebra  $  \mathfrak A \times \mathfrak B $
of  $  X \times Y $
containing all sets of the form  $  A \times B $, 
where  $  A \in \mathfrak A $
and  $  B \in \mathfrak B $. 
If  $  \Gamma $
is a measurable multi-valued mapping from  $  ( X , \mathfrak A ) $
to a complete separable metric space  $  ( Y , \mathfrak B ) $, 
where  $  \mathfrak B $
is the Borel  $  \sigma $-
algebra of  $  Y $, 
then  $  \Gamma $
has measurable sections  $  f $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''1–2''' , Acad. Press  (1966–1968)  (Translated from French)</TD></TR></table>

====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 [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Michael,   "Continuous selections"  ''Ann. of Math.'' , '''63'''  (1956)  pp. 361–382</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.A. Michael,   "A survey of continuous selections"  W.M. Fleischmann (ed.) , ''Set valued mappings, selections and topological properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524091.png" /> (Proc. Conf. SUNY Buffalo, 1969)'' , ''Lect. notes in math.'' , '''171''' , Springer  (1970)  pp. 54–58</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Przeslawski,   D. Yost,   "Continuity properties of selectors and Michael's Theorem"  ''Mich. Math. J.'' , '''36'''  (1989)  pp. 113–134</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Parthasarathy,   "Selection theorems and their applications" , ''Lect. notes in math.'' , '''263''' , Springer  (1972)</TD></TR></table>