Selection theorem

In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.[1]

## Preliminaries

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Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently,  F : X → P ( Y ) {\displaystyle F:X\rightarrow {\mathcal {P}}(Y)} is a function from X to the power set of Y.

A function  f : X → Y {\displaystyle f:X\rightarrow Y} is said to be a selection of F if

∀ x ∈ X : f ( x ) ∈ F ( x ) . {\displaystyle \forall x\in X:\,\,\,f(x)\in F(x)\,.}

In other words, given an input x for which the original function F returns multiple values, the new function f returns a single value. This is a special case of a choice function.

The axiom of choice implies that a selection function always exists; however, it is often important that the selection have some "nice" properties, such as continuity or measurability. This is where the selection theorems come into action: they guarantee that, if F satisfies certain properties, then it has a selection f that is continuous or has other desirable properties.

## Selection theorems for set-valued functions

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The Michael selection theorem[2] says that the following conditions are sufficient for the existence of a continuous selection:

* X is a paracompact space;
* Y is a Banach space;
* F is lower hemicontinuous;
* for all x in X, the set F(x) is nonempty, convex and closed.

The approximate selection theorem[3] states the following:

> Suppose X is a compact metric space, Y a non-empty compact, convex subset of a normed vector space, and Φ: X → P ( Y ) {\displaystyle {\mathcal {P}}(Y)} a multifunction all of whose values are compact and convex. If graph(Φ) is closed, then for every ε > 0 there exists a continuous function f : X → Y with graph(f) ⊂ [graph(Φ)]ε.

Here,  [ S ] ε {\displaystyle [S]_{\varepsilon }} denotes the  ε {\displaystyle \varepsilon } -dilation of  S {\displaystyle S} , that is, the union of radius- ε {\displaystyle \varepsilon } open balls centered on points in  S {\displaystyle S} . The theorem implies the existence of a continuous approximate selection.

Another set of sufficient conditions for the existence of a continuous approximate selection is given by the Deutsch–Kenderov theorem,[4] whose conditions are more general than those of Michael's theorem (and thus the selection is only approximate):

* X is a paracompact space;
* Y is a normed vector space;
* F is almost lower hemicontinuous, that is, at each  x ∈ X {\displaystyle x\in X} , for each neighborhood  V {\displaystyle V} of  0 {\displaystyle 0} there exists a neighborhood  U {\displaystyle U} of  x {\displaystyle x} such that  ⋂ u ∈ U { F ( u ) + V } ≠ ∅ {\textstyle \bigcap _{u\in U}\\{F(u)+V\\}\neq \emptyset } ;
* for all x in X, the set F(x) is nonempty and convex.

In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if  Y {\displaystyle Y} is a locally convex topological vector space.[5]

The Yannelis-Prabhakar selection theorem[6] says that the following conditions are sufficient for the existence of a continuous selection:

* X is a paracompact Hausdorff space;
* Y is a linear topological space;
* for all x in X, the set F(x) is nonempty and convex;
* for all y in Y, the inverse set F−1(y) is an open set in X.

The Kuratowski and Ryll-Nardzewski measurable selection theorem says that if X is a Polish space and  B {\displaystyle {\mathcal {B}}} its Borel σ-algebra,  C l ( X ) {\displaystyle \mathrm {Cl} (X)} is the set of nonempty closed subsets of X,  ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} is a measurable space, and  F : Ω → C l ( X ) {\displaystyle F:\Omega \to \mathrm {Cl} (X)} is an  F {\displaystyle {\mathcal {F}}} -weakly measurable map (that is, for every open subset  U ⊆ X {\displaystyle U\subseteq X} we have  { ω ∈ Ω : F ( ω ) ∩ U ≠ ∅ } ∈ F {\displaystyle \\{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \\}\in {\mathcal {F}}} ), then  F {\displaystyle F} has a selection that is  ( F , B ) {\displaystyle ({\mathcal {F}},{\mathcal {B}})} -measurable.[7]

Other selection theorems for set-valued functions include:

* Bressan–Colombo directionally continuous selection theorem
* Castaing representation theorem
* Fryszkowski decomposable map selection
* Helly's selection theorem
* Zero-dimensional Michael selection theorem
* Robert Aumann measurable selection theorem

## Selection theorems for set-valued sequences

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* Blaschke selection theorem
* Maximum theorem

## References

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1. ^ Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9.
2. ^ Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics. Second Series. 63 (2): 361–382\. doi:10.2307/1969615. hdl:10338.dmlcz/119700. JSTOR 1969615. MR 0077107.
3. ^ Shapiro, Joel H. (2016). A Fixed-Point Farrago. Springer International Publishing. pp. 68–70\. ISBN 978-3-319-27978-7. OCLC 984777840.
4. ^ Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections". SIAM Journal on Mathematical Analysis. 14 (1): 185–194\. doi:10.1137/0514015.
5. ^ Xu, Yuguang (December 2001). "A Note on a Continuous Approximate Selection Theorem". Journal of Approximation Theory. 113 (2): 324–325\. doi:10.1006/jath.2001.3622.
6. ^ Yannelis, Nicholas C.; Prabhakar, N. D. (1983-12-01). "Existence of maximal elements and equilibria in linear topological spaces". Journal of Mathematical Economics. 12 (3): 233–245\. CiteSeerX 10.1.1.702.2938. doi:10.1016/0304-4068(83)90041-1. ISSN 0304-4068.
7. ^ V. I. Bogachev, "Measure Theory" Volume II, page 36.

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