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''point-to-set mapping''
 
''point-to-set mapping''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652401.png" /> associating with each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652402.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652403.png" /> a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652404.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652405.png" />. If for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652406.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652407.png" /> consists of one element, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652408.png" /> is called single-valued. A multi-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m0652409.png" /> can be treated as a single-valued mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524010.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524011.png" />, that is, into the set of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524012.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524014.png" />, their inclusion is naturally defined: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524015.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524017.png" />. For any family of multi-valued mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524019.png" />, the union and intersection are defined: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524025.png" />. For any family of multi-valued mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524027.png" />, the multi-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524028.png" /> is called the Cartesian product of the multi-valued mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524029.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524030.png" />. A section of a multi-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524031.png" /> is a single-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524034.png" />. The graph of a multi-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524035.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524036.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524037.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524038.png" /> into a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524039.png" /> is called upper semi-continuous if for every open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524040.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524041.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524042.png" />, or equivalently: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524043.png" /> and any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524045.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524049.png" />. A multi-valued mapping from a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524050.png" /> to a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524051.png" /> is called lower semi-continuous if for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524052.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524053.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524054.png" />. If a multi-valued mapping satisfies both properties simultaneously, then it is called a continuous multi-valued mapping.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524055.png" /> be a topological vector space. A multi-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524056.png" /> is called convex-compact valued if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524057.png" /> is a convex compact set for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524058.png" />. For a finite set of multi-valued mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524060.png" />, the algebraic sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524061.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524062.png" />. 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 $  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524063.png" /> be a [[Paracompact space|paracompact space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524064.png" /> a locally convex metric linear space (cf. [[Locally convex space|Locally convex space]]; [[Linear space|Linear space]]; [[Metric space|Metric space]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524065.png" /> be a multi-valued mapping which is upper semi-continuous and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524066.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524067.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524068.png" />. Then the multi-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524069.png" /> admits continuous sections. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524071.png" /> be spaces with given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524072.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524074.png" />; a multi-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524075.png" /> is called measurable if the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524076.png" /> belongs to the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524077.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524078.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524079.png" /> containing all sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524080.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524083.png" /> is a measurable multi-valued mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524084.png" /> to a complete separable metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524086.png" /> is the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524087.png" />-algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524088.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524089.png" /> has measurable sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065240/m06524090.png" />.
+
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====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:01, 6 June 2020


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 and $ Y $ a locally convex metric linear space (cf. Locally convex space; Linear 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

[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 (Proc. Conf. SUNY Buffalo, 1969) , Lect. notes in math. , 171 , Springer (1970) pp. 54–58
[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)
How to Cite This Entry:
Multi-valued mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-valued_mapping&oldid=47923
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article