Selection theorem

Given two sets X and Y, let F be a multivalued map 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 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.

1. 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.

2. The Deutsch–Kenderov theorem[3] 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 ${\displaystyle x\in X}$, for each neighborhood ${\displaystyle V}$ of ${\displaystyle 0}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ such that ${\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .}$
• For all x in X, the set F(x) is nonempty and convex.

These conditions guarantee that ${\displaystyle F}$ has continuous approximate selection, that is, for each neighborhood ${\displaystyle V}$ of ${\displaystyle 0}$ in ${\displaystyle Y}$ there is a continuous function ${\displaystyle f\colon X\mapsto Y}$ such that for each ${\displaystyle x\in X}$, ${\displaystyle f(x)\in F(X)+V}$.[3]

In a later note, Xu proved that Deutsch–Kenderov theorem is also valid if ${\displaystyle Y}$ is a locally convex topological vector space.[4]

3. The Yannelis-Prabhakar selection theorem[5] 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:

• ${\displaystyle X}$ is a Polish space and ${\displaystyle {\mathcal {B}}}$ its Borel σ-algebra;
• ${\displaystyle Cl(X)}$ is the set of nonempty closed subsets of ${\displaystyle X}$.
• ${\displaystyle (\Omega ,{\mathcal {F}})}$ a measurable space, and ${\displaystyle F:\Omega \to Cl(X)}$ a ${\displaystyle {\mathcal {F}}}$-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.[6]

List of selection theorems