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.

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