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]

## Contents

* 1 Preliminaries
* 2 Selection theorems for set-valued functions
* 3 Selection theorems for set-valued sequences
* 4 References

## Preliminaries

Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently, F:X->𝒫(Y) is a function from X to the power set of Y.

A function f:X->Y is said to be a selection of F if

∀x∈X:f(x)∈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

The approximate selection theorem[2] 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)

> 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

denotes the

-dilation of

, that is, the union of radius-

open balls centered on points in

. The theorem implies the existence of a continuous approximate selection.

The Michael selection theorem[3] 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.

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:

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

These conditions guarantee that F has a continuous approximate selection. This conclusion is thus weaker than in Michael's theorem.

In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if 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 its Borel σ-algebra, Cl(X) is the set of nonempty closed subsets of X, (Ω,ℱ) is a measurable space, and F:Ω->Cl(X) is an ℱ-weakly measurable map (that is, for every open subset U⊆X we have {ω∈Ω:F(ω)∩U≠∅}∈ℱ), then F has a selection that is (ℱ,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

* Blaschke selection theorem

## References

1. ↑ Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9.
2. ↑ Shapiro, Joel H. (2016). A Fixed-Point Farrago. Springer International Publishing. pp. 68–70. ISBN 978-3-319-27978-7. OCLC 984777840. http://worldcat.org/oclc/984777840.
3. ↑ Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics. Second Series 63 (2): 361–382. doi:10.2307/1969615.
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. 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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