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, $F:X\rightarrow\mathcal(Y)$ 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 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:

* ''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 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 →'' $\mathcal P(Y)$ 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'') ⊂ raph(Φ)sub>ε.

Here, \varepsilon 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):

* ''X'' is a paracompact space;
* ''Y'' is a normed vector space;
* ''F'' is ''almost lower hemicontinuous'', that is, at each for each neighborhood $V$ of $0$ there exists a neighborhood $U$ of $x$ such that
* for all ''x'' in ''X'', the set ''F''(''x'') is nonempty and convex.

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:

* ''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 $\mathcal B$ its Borel σ-algebra, $\mathrm(X)$ is the set of nonempty closed subsets of ''X'', $(\Omega, \mathcal F)$ is a measurable space, and $F : \Omega \to \mathrm(X)$ is an measurable map (that is, for every open subset $U \subseteq X$ we have then $F$ has a selection that is V. I. Bogachev
"Measure Theory"
Volume II, page 36.

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

Selection theorems for set-valued sequences

* Blaschke selection theorem
* Maximum theorem

References

{{Functional analysis

Theorems in functional analysis