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] Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently,
is said to be a selection of F if 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[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 →
a multifunction all of whose values are compact and convex.
open balls centered on points in
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): In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if
is a locally convex topological vector space.
is the set of nonempty closed subsets of X,
-weakly measurable map (that is, for every open subset
[7] Other selection theorems for set-valued functions include: