- Source: Kuratowski and Ryll-Nardzewski measurable selection theorem
In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.
Many classical selection results follow from this theorem and it is widely used in mathematical economics and optimal control.
Statement of the theorem
Let
X
{\displaystyle X}
be a Polish space,
B
(
X
)
{\displaystyle {\mathcal {B}}(X)}
the Borel σ-algebra of
X
{\displaystyle X}
,
(
Ω
,
F
)
{\displaystyle (\Omega ,{\mathcal {F}})}
a measurable space and
ψ
{\displaystyle \psi }
a multifunction on
Ω
{\displaystyle \Omega }
taking values in the set of nonempty closed subsets of
X
{\displaystyle X}
.
Suppose that
ψ
{\displaystyle \psi }
is
F
{\displaystyle {\mathcal {F}}}
-weakly measurable, that is, for every open subset
U
{\displaystyle U}
of
X
{\displaystyle X}
, we have
{
ω
:
ψ
(
ω
)
∩
U
≠
∅
}
∈
F
.
{\displaystyle \{\omega :\psi (\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}.}
Then
ψ
{\displaystyle \psi }
has a selection that is
F
{\displaystyle {\mathcal {F}}}
-
B
(
X
)
{\displaystyle {\mathcal {B}}(X)}
-measurable.
See also
Selection theorem
References
Kata Kunci Pencarian:
- Kuratowski and Ryll-Nardzewski measurable selection theorem
- Ryll-Nardzewski theorem
- Kazimierz Kuratowski
- Czesław Ryll-Nardzewski
- Selection theorem
- List of things named after Kazimierz Kuratowski
- Set-valued function
