# 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 ⋂$_{u∈U}${F(u)+V}≠∅{\textstyle \bigcap _{u\in U}\{F(u)+V\}\neq \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 $\{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}$), 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