In measure theory, or at least in the approach to it via the domain theory, a valuation is a map from the class of open sets of a topological space to the set positive real numbers including infinity. It is a concept closely related to that of a measure, and as such, it finds applications in measure theory, probability theory, and theoretical computer science.
Domain/Measure theory definition
Let
(
X
,
T
)
be a topological space: a valuation is any map
v
:
T
→
R
+
∪
{
+
∞
}
satisfying the following three properties
v
(
∅
)
=
0
Strictness property
v
(
U
)
≤
v
(
V
)
if
U
⊆
V
U
,
V
∈
T
Monotonicity property
v
(
U
∪
V
)
+
v
(
U
∩
V
)
=
v
(
U
)
+
v
(
V
)
∀
U
,
V
∈
T
Modularity property
The definition immediately shows the relationship between a valuation and a measure: the properties of the two mathematical object are often very similar if not identical, the only difference being that the domain of a measure is the Borel algebra of the given topological space, while the domain of a valuation is the class of open sets. Further details and references can be found in Alvarez-Manilla et al. 2000 and Goulbault-Larrecq 2002.
A valuation (as defined in domain theory/measure theory) is said to be continuous if for every directed family
{
U
i
}
i
∈
I
of open sets (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes
i
and
j
belonging to the index set
I
, there exists an index
k
such that
U
i
⊆
U
k
and
U
j
⊆
U
k
) the following equality holds:
v
(
⋃
i
∈
I
U
i
)
=
sup
i
∈
I
v
(
U
i
)
.
A valuation (as defined in domain theory/measure theory) is said to be simple if it is a finite linear combination with non-negative coefficients of Dirac valuations, i.e.
v
(
U
)
=
∑
i
=
1
n
a
i
δ
x
i
(
U
)
∀
U
∈
T
where
a
i
is always greather than or at least equal to zero for all index
i
. Simple valuations are obviously continuous in the above sense. The supremum of a directed family of simple valuations (i.e. an indexed family of simple valuations which is also directed in the sense that for each pair of indexes
i
and
j
belonging to the index set
I
, there exists an index
k
such that
v
i
(
U
)
≤
v
k
(
U
)
and
v
j
(
U
)
⊆
v
k
(
U
)
) is called quasi-simple valuation
v
¯
(
U
)
=
sup
i
∈
I
v
i
(
U
)
∀
U
∈
T
.
The extension problem for a given valuation (in the sense of domain theory/measure theory) consists in finding under what type of conditions it can be extended to a measure on a proper topological space, which may or may not be the same space where it is defined: the papers Alvarez-Manilla, Edalat & Saheb-Djahromi 2000 and Goulbault-Larrecq 2002 in the reference section are devoted to this aim and give also several historical details.
The concepts of valuation on convex sets and valuation on manifolds are a generalization of valuation in the sense of domain/measure theory. A valuation on convex sets is allowed to assume complex values, and the underlying topological space is the set of non-empty convex compact subsets of a finite-dimensional vector space: a valuation on manifolds is a complex valued finitely additive measure defined on a proper subset of the class of all compact submanifolds of the given manifolds. Details can be found in several arxiv papers of prof. Semyon Alesker.
Let
(
X
,
T
)
be a topological space, and let
x
be a point of
X
: the map
δ
x
(
U
)
=
{
0
if
x
∉
U
1
if
x
∈
U
∀
U
∈
T
is a valuation in the domain theory/measure theory, sense called Dirac valuation. This concept bears its origin from distribution theory as it is an obvious transposition to valuation theory of Dirac distribution: as seen above, Dirac valuations are the "bricks" simple valuations are made of.
