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.

Preliminaries

Given two sets X and Y, let F be a multivalued map 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 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 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 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 $x\in X$, for each neighborhood $V$ of $0$ there exists a neighborhood $U$ of $x$ such that $\cap _{u\in U}\{F(u)+V\}\neq \emptyset .$
• For all x in X, the set F(x) is nonempty and convex.

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

In a later note, Xu proved that Deutsch–Kenderov theorem is also valid if $Y$ is a locally convex topological vector space.

3. 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.

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 and ${\mathcal {B}}$ its Borel σ-algebra;
• $Cl(X)$ is the set of nonempty closed subsets of $X$.
• $(\Omega ,{\mathcal {F}})$ a measurable space, and $F:\Omega \to Cl(X)$ a ${\mathcal {F}}$-weakly measurable map (that is, for every open subset $U\subseteq X$ we have $\{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}$).

Then $F$ has a selection that is $({\mathcal {F}},{\mathcal {B}})$-measurable.

See also

List of selection theorems