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.^[1]

Preliminaries

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

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

[math]\displaystyle{ \forall x \in X: \,\,\, f(x) \in F(x) \,. }[/math]

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 approximate selection theorem^[2] 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 → [math]\displaystyle{ \mathcal P(Y) }[/math] 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, [math]\displaystyle{ [S]_\varepsilon }[/math] denotes the [math]\displaystyle{ \varepsilon }[/math]-dilation of [math]\displaystyle{ S }[/math], that is, the union of radius-[math]\displaystyle{ \varepsilon }[/math] open balls centered on points in [math]\displaystyle{ S }[/math]. The theorem implies the existence of a continuous approximate selection.

The Michael selection theorem^[3] 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.

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:

• X is a paracompact space;
• Y is a normed vector space;
• F is almost lower hemicontinuous, that is, at each [math]\displaystyle{ x \in X }[/math], for each neighborhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ 0 }[/math] there exists a neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x }[/math] such that [math]\displaystyle{ \bigcap_{u \in U} \{F(u)+V\} \ne \emptyset }[/math];
• for all x in X, the set F(x) is nonempty and convex.

These conditions guarantee that [math]\displaystyle{ F }[/math] has a continuous approximate selection. This conclusion is thus weaker than in Michael's theorem.

In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if [math]\displaystyle{ Y }[/math] 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 [math]\displaystyle{ \mathcal B }[/math] its Borel σ-algebra, [math]\displaystyle{ \mathrm{Cl}(X) }[/math] is the set of nonempty closed subsets of X, [math]\displaystyle{ (\Omega, \mathcal F) }[/math] is a measurable space, and [math]\displaystyle{ F : \Omega \to \mathrm{Cl}(X) }[/math] is an [math]\displaystyle{ \mathcal F }[/math]-weakly measurable map (that is, for every open subset [math]\displaystyle{ U \subseteq X }[/math] we have [math]\displaystyle{ \{\omega \in \Omega : F(\omega) \cap U \neq \empty \} \in \mathcal F }[/math]), then [math]\displaystyle{ F }[/math] has a selection that is [math]\displaystyle{ (\mathcal F, \mathcal B) }[/math]-measurable.^[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

References

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