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(Φ) is closed, then for every ε > 0 there exists a continuous function f : X → Y with graph(f) ⊂ [graph(Φ)]ε."Here, denotes the \varepsilon-dilation of S, that is, the union of radius-\varepsilon open balls centered on points in S. 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 (and thus the selection is only approximate): In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if Y 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 \mathcal B its Borel σ-algebra, is the set of nonempty closed subsets of X, is a measurable space, and is an \mathcal F-weakly measurable map (that is, for every open subset we have ), then F has a selection that is -measurable. Other selection theorems for set-valued functions include:

Selection theorems for set-valued sequences