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.{{cite book|title=Fixed Point Theorems with Applications to Economics and Game Theory|last=Border|first=Kim C.|publisher=Cambridge University Press|year=1989|isbn=0-521-26564-9}}

Preliminaries

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

A function $f: X \rightarrow Y$ is said to be a selection of F if

: $\forall x \in X: \,\,\, f(x) \in F(x) \,.$

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

The Michael selection theorem{{Cite journal|last1=Michael|first1=Ernest|year=1956|title=Continuous selections. I|journal=Annals of Mathematics|series=Second Series|volume=63|issue=2|pages=361–382|doi=10.2307/1969615|jstor=1969615|mr=0077107|author1-link=Ernest Michael|hdl=10338.dmlcz/119700|hdl-access=free}} 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{{Cite book |last=Shapiro |first=Joel H. |url=http://worldcat.org/oclc/984777840 |title=A Fixed-Point Farrago |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-27978-7 |pages=68–70 |oclc=984777840}} 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 → $\mathcal P(Y)$ 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]_\varepsilon

denotes the

\varepsilon

-dilation of

S

, that is, the union of radius-

\varepsilon

open balls centered on points in

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,{{cite journal|last1=Deutsch|first1=Frank|last2=Kenderov|first2=Petar|date=January 1983|title=Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections|journal=SIAM Journal on Mathematical Analysis|volume=14|issue=1|pages=185–194|doi=10.1137/0514015}} 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 {{nowrap| $x \in X$ ,}} for each neighborhood $V$ of $0$ there exists a neighborhood $U$ of $x$ such that {{nowrap| $\bigcap_{u \in U} \{F(u)+V\} \ne \emptyset$ ;}}
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 $Y$ is a locally convex topological vector space.{{cite journal|last1=Xu|first1=Yuguang|date=December 2001|title=A Note on a Continuous Approximate Selection Theorem|journal=Journal of Approximation Theory|volume=113|issue=2|pages=324–325|doi=10.1006/jath.2001.3622|doi-access=free}}

The Yannelis-Prabhakar selection theorem{{Cite journal|last=Yannelis|first=Nicholas C.|last2=Prabhakar|first2=N. D.|date=1983-12-01|title=Existence of maximal elements and equilibria in linear topological spaces|journal=Journal of Mathematical Economics|volume=12|issue=3|pages=233–245|doi=10.1016/0304-4068(83)90041-1|issn=0304-4068|citeseerx=10.1.1.702.2938}} 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 $\mathcal B$ its Borel σ-algebra, $\mathrm{Cl}(X)$ is the set of nonempty closed subsets of X, $(\Omega, \mathcal F)$ is a measurable space, and $F : \Omega \to \mathrm{Cl}(X)$ is an {{nowrap| $\mathcal F$ -weakly}} measurable map (that is, for every open subset $U \subseteq X$ we have {{nowrap| $\{\omega \in \Omega : F(\omega) \cap U \neq \empty \} \in \mathcal F$ ),}} then $F$ has a selection that is {{nowrap| $(\mathcal F, \mathcal B)$ -measurable.}}V. I. Bogachev, [https://www.springer.com/math/analysis/book/978-3-540-34513-8 "Measure Theory"] Volume II, page 36.

Other selection theorems for set-valued functions include:

Selection theorems for set-valued sequences

References

Category:Theorems in functional analysis