# 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 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 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:

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 ${\textstyle \bigcap _{u\in U}\{F(u)+V\}\neq \emptyset }$ ;
• for all x in X, the set F(x) is nonempty and convex.

These conditions guarantee that $F$  has a 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 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:

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 $\{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}$ ), then $F$  has a selection that is $({\mathcal {F}},{\mathcal {B}})$ -measurable.

Other selection theorems for set-valued functions include: