• Source: 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
→


P


(
Y
)


{\displaystyle F:X\rightarrow {\mathcal {P}}(Y)}

is a function from X to the power set of Y.
A function



f
:
X
→
Y


{\displaystyle f:X\rightarrow Y}

is said to be a selection of F if




∀
x
∈
X
:



f
(
x
)
∈
F
(
x
)

.


{\displaystyle \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 →





P


(
Y
)


{\displaystyle {\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) ⊂ [graph(Φ)]ε.Here,



[
S

]

ε




{\displaystyle [S]_{\varepsilon }}

denotes the



ε


{\displaystyle \varepsilon }

-dilation of



S


{\displaystyle S}

, that is, the union of radius-



ε


{\displaystyle \varepsilon }

open balls centered on points in



S


{\displaystyle 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



x
∈
X


{\displaystyle x\in X}

, for each neighborhood



V


{\displaystyle V}

of



0


{\displaystyle 0}

there exists a neighborhood



U


{\displaystyle U}

of



x


{\displaystyle x}

such that




⋂

u
∈
U


{
F
(
u
)
+
V
}
≠
∅


{\textstyle \bigcap _{u\in U}\{F(u)+V\}\neq \emptyset }

;
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


{\displaystyle 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





B




{\displaystyle {\mathcal {B}}}

its Borel σ-algebra,




C
l

(
X
)


{\displaystyle \mathrm {Cl} (X)}

is the set of nonempty closed subsets of X,



(
Ω
,


F


)


{\displaystyle (\Omega ,{\mathcal {F}})}

is a measurable space, and



F
:
Ω
→

C
l

(
X
)


{\displaystyle F:\Omega \to \mathrm {Cl} (X)}

is an





F




{\displaystyle {\mathcal {F}}}

-weakly measurable map (that is, for every open subset



U
⊆
X


{\displaystyle U\subseteq X}

we have



{
ω
∈
Ω
:
F
(
ω
)
∩
U
≠
∅
}
∈


F




{\displaystyle \{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}}

), then



F


{\displaystyle F}

has a selection that is



(


F


,


B


)


{\displaystyle ({\mathcal {F}},{\mathcal {B}})}

-measurable.
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

Kata Kunci Pencarian:

• Bilangan prima
• Statistika
• George R. Price
• Variabel acak
• Teorema Bohr–Mollerup
• Statistika matematika
• Ilmu aktuaria
• Metode simpleks
• James Clerk Maxwell
• Model generatif
• Selection theorem
• Rellich–Kondrachov theorem
• Helly's selection theorem
• Fisher's fundamental theorem of natural selection
• Mahler's compactness theorem
• Kuratowski and Ryll-Nardzewski measurable selection theorem
• Blaschke selection theorem
• Czesław Ryll-Nardzewski
• List of theorems
• Fraňková–Helly selection theorem