Selection theorem

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.

The approximate selection theorem[2] says that the following conditions are sufficient for the existence of a continuous selection:

• ${\displaystyle X}$: compact metric space

• ${\displaystyle Y}$: nonempty compact, convex subset of a normed linear space

• ${\displaystyle F:X\to 2^{Y}}$ a set-valued function, all values nonempty, compact, convex.

• ${\displaystyle F}$ has closed graph.

• For every ${\displaystyle \varepsilon >0}$ there exists a continuous function ${\displaystyle f:X\rightarrow Y}$ with ${\displaystyle \operatorname {graph} (f)\subset [\operatorname {graph} (F)]_{\varepsilon }}$, where ${\displaystyle [S]_{\epsilon }}$ is the ${\displaystyle \epsilon }$-dilation of ${\displaystyle S}$, that is, the union of radius-${\displaystyle \epsilon }$ open balls centered on points in ${\displaystyle S}$.

The Michael selection theorem[3] 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 Deutsch–Kenderov theorem[4] 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 ${\textstyle \bigcap _{u\in U}\{F(u)+V\}\neq \emptyset }$;
• for all x in X, the set F(x) is nonempty and convex.

These conditions guarantee that ${\displaystyle F}$ has a 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 ${\displaystyle x\in X}$, ${\displaystyle f(x)\in F(X)+V}$.[4]

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

• Blaschke selection theorem