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This page is about valuation in measure theory. For valuation in algebra (on rings/fields) see at valuation.

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Contents

Idea

A valuation is a construction analogous to that of a measure, which is however more compatible with constructive mathematics, and readily generalizable to contexts such as point-free topology.

Definition

Valuations on lattices

Let $L$ be a distributive lattice with a bottom element $\bottom$. A valuation or evaluation on $L$ is a map $\nu$ from $L$ into the space of non-negative lower reals, with the following properties:

• Monotonicity: for all $x,y$ in $L$, $x\le y$ implies $\nu(x)\le\nu(y)$;

• Strictness (or unitality): $\nu(\bottom)=0$;

• Modularity: for all $x,y$ in $L$,

Moreover, we call a valuation continuous if the following property holds, which is an instance of Scott continuity, as well as of $\tau$-additivity:

• Continuity: for every directed net $\{x_\lambda\}_{\lambda\in\Lambda}$ in $L$ admitting a supremum,

For now, see more in Vickers.

Valuations on locales and topological spaces

Let $L$ be a locale. Then a valuation on $L$ is by definition a valuation on its frame $\mathcal{O}(L)$. Similarly, a valuation on a topological space is a valuation on the lattice of its open sets.

Valuations on locales are used in the topos approach to quantum mechanics and the Bohr topos.

Examples

Dirac valuation

Let $X$ be a topological space, and let $x\in X$ be a point. The Dirac valuation at $x$, which we denote by $\delta_x$, maps an open set $U\subseteq X$ to $1$ if $x\in U$, and to $0$ otherwise.

Simple valuations

On a topological space, a simple valuation is a finite convex (or linear) combination of Dirac valuations, i.e. a valuation in the form

for $x_i\in X$, and for positive (lower) real numbers $\alpha_i$, possibly summing to one.

Borel measures

For more on this, see $\tau$-additive measure.

Let $X$ be a topological space, and let $\mu$ be a measure defined on the Borel $\sigma$-algebra of $X$. Then the restriction of $\mu$ to the open subsets of $X$ is a valuation. The valuation is continuous if and only if $\mu$ is $\tau$-additive.

The converse problem of whether a valuation is the restriction of a Borel measure is more difficult, see below.

Measure-theoretic structures and properties

Null sets and support

Valuations admit a notion of support to that of measures. In particular, continuous valuations, just as $\tau$-additive measures, have a well-defined and well-behaved support.

Let $\nu$ be a valuation on a locale or topological space $X$, and $U$ an open set of $X$ (i.e. an element of the corresponding frame). We say that $U$ is a null or measure zero set for $\nu$ if $\nu(U)=0$. The complement of $U$, which is a closed subspace of $X$, is said to have full measure.

Since a finite union of null sets is null, null sets form a directed net in the frame. Therefore, if $\nu$ is a continuous valuation, it admits a unique maximal null open set. The complement of this set, which is the largest closed subspace of full measure, is called the support of $\nu$.

Integration

(…)

Extending valuations to measures

As we have seen above, a Borel measure always restricts to a valuation. It is natural to ask the converse question of whether a valuation can always be extended to a Borel measure. In general, the answer is negative. In the case of continuous valuations, however, one would expect that in many cases the valuation can be extended to a $\tau$-additive Borel measure.

The question is known, for example, to be true on all regular Hausdorff ($T_3$) spaces:

Theorem (see Manilla, Theorems 3.23 and 3.27). On every $T_3$ topological space, and on every locally compact sober space, a locally finite continuous valuation extends uniquely to a regular, $\tau$-additive Borel measure.

This includes in particular every metric space, and every compact Hausdorff space. So, in many spaces of interest for analysis and probability theory, working with measures and working with valuations is only a difference in the language.

The more general question of whether one can extend a finite continuous valuation to a Borel measure on any sober space, at the present time, is still open.

References

For a general treatment, see

• Steve Vickers, A monad of valuation locales, 2011. Link here.

• Wikipedia, Valuation (measure theory)

For the theory of integration over valuations, see

• Steve Vickers, A localic theory of lower and upper integrals, 2008. Link here.

• Thierry Coquand and Bas Spitters, Integrals and Valuations, 2009. Link here.

For the problem of extending valuations to measures, see

• Mauricio Alvarez Manilla, Abbas Edalat, and Nasser Saheb-Djahromi, An extension result for continuous valuations, 1998. Link here.

• Mauricio Alvarez Manilla, Measure theoretic results for continuous valuations on partially ordered spaces, Dissertation, 2000. Link here.

• Klaus Keimel and Jimmie D. Lawson, Measure extension theorems for $T_0$ spaces, 2004. Link here.