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, ${\displaystyle F:X\rightarrow {\mathcal {P}}(Y)}$ is a function from X to the power set of Y.

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

${\displaystyle \forall x\in X:\,\,\,f(x)\in F(x)\,.}$

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^[2] 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^[3] 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 → ${\displaystyle {\mathcal {P}}(Y)}$ 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, ${\displaystyle [S]_{\varepsilon }}$ denotes the ${\displaystyle \varepsilon }$-dilation of ${\displaystyle S}$, that is, the union of radius-${\displaystyle \varepsilon }$ open balls centered on points in ${\displaystyle S}$. 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,^[4] 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 ${\displaystyle x\in X}$, for each neighborhood ${\displaystyle V}$ of ${\displaystyle 0}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ 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.

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

References