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.

Preliminaries

Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently, is a function from X to the power set of Y.
A function 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.

Selection theorems for set-valued functions

The Michael selection theorem says that the following conditions are sufficient for the existence of a continuous selection:
The approximate selection theorem 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. If graph is closed, then for every ε > 0 there exists a continuous function f : X → Y with graph ⊂ ε.
Here, denotes the -dilation of, that is, the union of radius- 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, whose conditions are more general than those of Michael's theorem :
  • X is a paracompact space;
  • Y is a normed vector space;
  • F is almost lower hemicontinuous, that is, at each for each neighborhood of there exists a neighborhood of such that
  • for all x in X, the set F is nonempty and convex.
In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if is a locally convex topological vector space.
The Yannelis-Prabhakar selection theorem says that the following conditions are sufficient for the existence of a continuous selection:
The Kuratowski and Ryll-Nardzewski measurable selection theorem says that if X is a Polish space and its Borel σ-algebra, is the set of nonempty closed subsets of X, is a measurable space, and is an measurable map (that is, for every open subset we have then has a selection that is
Other selection theorems for set-valued functions include: