Selection theorem

In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of 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 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.

Selection theorems for set-valued functions

• Bressan–Colombo directionally continuous selection theorem
• Castaing representation theorem
• Fryszkowski decomposable map selection
• Helly's selection theorem
• Kuratowski and Ryll-Nardzewski measurable selection theorem
• Michael selection theorem
• Zero-dimensional Michael selection theorem
• Robert Aumann measurable selection theorem
• Yannelis-Prabhakar selection theorem

Selection theorems for set-valued sequences

• Blaschke selection theorem