Selection theorem explained

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

Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently,

F:X → l{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\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^[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 →

l 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,^[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 for each neighborhood

V of 0 there exists a neighborhood U of x such that

• 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.^[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

l B its Borel σ-algebra, Cl (X) is the set of nonempty closed subsets of X, (\Omega, l F) is a measurable space, and F : \Omega \to Cl (X) is an measurable map (that is, for every open subset U \subseteq X we have then F has a selection that is ^[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
• Maximum theorem

References

1. Book: Border, Kim C.. Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. 1989. 0-521-26564-9.
2. Michael. Ernest. 1956. Continuous selections. I. Annals of Mathematics. Second Series. 63. 2. 361–382. 10.2307/1969615. 1969615. 0077107. Ernest Michael. 10338.dmlcz/119700. free.
3. Book: Shapiro, Joel H. . A Fixed-Point Farrago . 2016 . Springer International Publishing . 978-3-319-27978-7 . 68–70 . 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. 10.1137/0514015.
5. Xu. Yuguang. December 2001. A Note on a Continuous Approximate Selection Theorem. Journal of Approximation Theory. 113. 2. 324–325. 10.1006/jath.2001.3622. free.
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. 10.1016/0304-4068(83)90041-1. 0304-4068. 10.1.1.702.2938.
7. V. I. Bogachev, "Measure Theory" Volume II, page 36.