Selection theorem

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 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 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, 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 x \in X, for each neighborhood V of 0 there exists a neighborhood U of x such that \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.

The Yannelis-Prabhakar selection theorem 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 \mathcal F-weakly measurable map (that is, for every open subset U \subseteq X we have {\omega \in \Omega : F(\omega) \cap U \neq \empty } \in \mathcal F), then F has a selection that is (\mathcal F, \mathcal B)-measurable.

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

References

1. Border, Kim C.. (1989). "Fixed Point Theorems with Applications to Economics and Game Theory". Cambridge University Press.
2. (1956). "Continuous selections. I". [[Annals of Mathematics]].
3. Shapiro, Joel H.. (2016). "A Fixed-Point Farrago". Springer International Publishing.
4. (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections". SIAM Journal on Mathematical Analysis.
5. (December 2001). "A Note on a Continuous Approximate Selection Theorem". Journal of Approximation Theory.
6. Yannelis, Nicholas C.. (1983-12-01). "Existence of maximal elements and equilibria in linear topological spaces". Journal of Mathematical Economics.
7. V. I. Bogachev, [https://www.springer.com/math/analysis/book/978-3-540-34513-8 "Measure Theory"] Volume II, page 36.