Selection theorem

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

Preliminaries

Given two sets X and Y, let F be a multivalued map 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 be continuous or measurable. 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

1. 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.

2. The Deutsch–Kenderov theorem[3] generalizes Michael's theorem as follows:

• 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 ${\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .}$
• For all x in X, the set F(x) is nonempty and convex.

These conditions guarantee that ${\displaystyle F}$ has continuous approximate selection, that is, for each neighborhood ${\displaystyle V}$ of ${\displaystyle 0}$ in ${\displaystyle Y}$ there is a continuous function ${\displaystyle f\colon X\mapsto Y}$ such that for each ${\displaystyle x\in X}$, ${\displaystyle f(x)\in F(X)+V}$.[3]

In a later note, Xu proved that Deutsch–Kenderov theorem is also valid if ${\displaystyle Y}$ is a locally convex topological vector space.[4]

3. The Yannelis-Prabhakar selection theorem[5] 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.

4. The Kuratowski and Ryll-Nardzewski measurable selection theorem says that the following conditions are sufficient for the existence of a measurable selection:

• X is a Polish space;
• Y is the set of nonempty closed subsets of X.
• ℬ(X) the Borel σ-algebra of X, (Ω, B) a measurable space, and ${\textstyle F:\Omega \to Y}$.
• F is B-weakly measurable, that is, for every open set U of X, we have

${\displaystyle \{\omega :F(\omega )\cap U\neq \emptyset \}\in B.}$

Then F has a selection that is B-ℬ(X)-measurable.[6]

See also

List of selection theorems

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. 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. 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.
4. 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.
5. 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.
6. V. I. Bogachev, "Measure Theory" Volume II, page 36.