nLab
valuation (measure theory)

Contents

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


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

  • Monotonicity: for all x,yx,y in LL, xyx\le y implies ν(x)ν(y)\nu(x)\le\nu(y);

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

  • Modularity: for all x,yx,y in LL,

ν(x)+ν(y)=ν(xy)+ν(xy). \nu(x) + \nu(y) = \nu(x \vee y) + \nu(x \wedge y) .

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

  • Continuity: for every directed net {x λ} λΛ\{x_\lambda\}_{\lambda\in\Lambda} in LL admitting a supremum,
ν(sup λx λ)=sup λν(x λ). \nu \big( \sup_{\lambda} x_\lambda \big) = \sup_\lambda \nu(x_\lambda) .

For now, see more in Vickers.

Valuations on locales and topological spaces

Let LL be a locale. Then a valuation on LL is by definition a valuation on its frame 𝒪(L)\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 XX be a topological space, and let xXx\in X be a point. The Dirac valuation at xx, which we denote by δ x\delta_x, maps an open set UXU\subseteq X to 11 if xUx\in U, and to 00 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

ν= i=1 nα iδ x i, \nu = \sum_{i=1}^n \alpha_i\,\delta_{x_i} ,

for x iXx_i\in X, and for positive (lower) real numbers α i\alpha_i, possibly summing to one.

Borel measures

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

Let XX be a topological space, and let μ\mu be a measure defined on the Borel σ \sigma -algebra of XX. Then the restriction of μ\mu to the open subsets of XX is a valuation. The valuation is continuous if and only if μ\mu is τ \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 τ \tau -additive measures, have a well-defined and well-behaved support.

Let ν\nu be a valuation on a locale or topological space XX, and UU an open set of XX (i.e. an element of the corresponding frame). We say that UU is a null or measure zero set for ν\nu if ν(U)=0\nu(U)=0. The complement of UU, which is a closed subspace of XX, 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 ν\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 ν\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 τ \tau -additive Borel measure.

The question is known, for example, to be true on all regular Hausdorff (T 3T_3) spaces:

Theorem (see Manilla, Theorems 3.23 and 3.27). On every T 3T_3 topological space, and on every locally compact sober space, a locally finite continuous valuation extends uniquely to a regular, τ \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.

However, we do have the following result for regular spaces.

Theorem (Theorem 4.4 in Manilla). Any locally finite continuous valuation on a regular topological space extends uniquely to a regular τ-smooth Borel measure.

References

For a general treatment, see

For the theory of integration over valuations, see

For the problem of extending valuations to measures, see

  • Mauricio Alvarez-Manilla, Extension of valuations on locally compact sober spaces.

  • 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 0T_0 spaces, 2004. Link here.

Last revised on June 27, 2019 at 15:21:47. See the history of this page for a list of all contributions to it.