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 multi-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.[1]

PreliminariesEdit

Given two sets X and Y, let F be a multivalued map 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 functionsEdit

The Michael selection theorem[2] says that the following conditions are sufficient for the existence of a continuous selection:

The Deutsch–Kenderov theorem[3] generalizes Michael's theorem as follows:

  • 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(x) is nonempty and convex.

These conditions guarantee that   has a continuous approximate selection, that is, for each neighborhood   of   in   there is a continuous function   such that for each  ,  .[3]

In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if   is a locally convex topological vector space.[4]

The Yannelis-Prabhakar selection theorem[5] 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  -weakly measurable map (that is, for every open subset   we have  ), then   has a selection that is  -measurable.[6]

Other selection theorems for set-valued functions include:

Selection theorems for set-valued sequencesEdit

ReferencesEdit

  1. ^ Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9.
  2. ^ Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics. Second Series. 63 (2): 361–382. doi:10.2307/1969615. hdl:10338.dmlcz/119700. JSTOR 1969615. MR 0077107.
  3. ^ a b Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections". SIAM Journal on Mathematical Analysis. 14 (1): 185–194. doi:10.1137/0514015.
  4. ^ Xu, Yuguang (December 2001). "A Note on a Continuous Approximate Selection Theorem". Journal of Approximation Theory. 113 (2): 324–325. doi:10.1006/jath.2001.3622.
  5. ^ Yannelis, Nicholas C.; Prabhakar, N. D. (1983-12-01). "Existence of maximal elements and equilibria in linear topological spaces". Journal of Mathematical Economics. 12 (3): 233–245. doi:10.1016/0304-4068(83)90041-1. ISSN 0304-4068.
  6. ^ V. I. Bogachev, "Measure Theory" Volume II, page 36.