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 (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)