Selection theorem

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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#edit

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

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

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

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#edit

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 → ${\displaystyle {\mathcal {P(Y)}$ 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, ${\displaystyle [S]_{\varepsilon$ denotes the ${\displaystyle \varepsilon }$ -dilation of ${\displaystyle S}$ , that is, the union of radius- ${\displaystyle \varepsilon }$ open balls centered on points in ${\displaystyle S}$ . 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 ${\displaystyle x\in X}$ , for each neighborhood ${\displaystyle V}$ of ${\displaystyle 0}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ such that ${\textstyle \bigcap _{u\in U}\{F(u)+V\}\neq \emptyset }$ ;
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 ${\displaystyle Y}$ 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 ${\displaystyle {\mathcal {B$ its Borel σ-algebra, ${\displaystyle \mathrm {Cl} (X)}$ is the set of nonempty closed subsets of X, ${\displaystyle (\Omega ,{\mathcal {F)}$ is a measurable space, and ${\displaystyle F:\Omega \to \mathrm {Cl} (X)}$ is an ${\displaystyle {\mathcal {F$ -weakly measurable map (that is, for every open subset ${\displaystyle U\subseteq X}$ we have ${\displaystyle \{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \}\in {\mathcal {F$ ), then ${\displaystyle F}$ has a selection that is ${\displaystyle ({\mathcal {F,{\mathcal {B)}$ -measurable.^[7]

Other selection theorems for set-valued functions include:

Selection theorems for set-valued sequences#edit

References#edit

^ Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9.
^ 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.
^ Shapiro, Joel H. (2016). A Fixed-Point Farrago. Springer International Publishing. pp. 68–70. ISBN 978-3-319-27978-7. OCLC 984777840.
^ 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.
^ 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.
^ 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.
^ V. I. Bogachev, "Measure Theory" Volume II, page 36.

Categories Theorems in functional analysis

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