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nLab valuation (measure theory) (Rev #6, changes)

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

Context

Measure and probability theory

measure theory

probability theory

(quantum probability)

Measure theory

Probability theory

Information geometry

Thermodynamics

Theorems

Applications

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:

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

  • StrictnessStrictness (or unitalityunitality): ν(⊥)=0\nu(\bottom)=0;

  • ModularityModularity: for all x,yx,y in LL,

ν(x)+ν(y)=ν(x∨y)+ν(x∧y). \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 $\tau$-smoothness $\tau$-additivity:

  • ContinuityContinuity: for every directed net {x λ} λ∈Λ\{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). In particular, 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 x∈Xx\in X be a point. The Dirac valuation at xx , maps which an we open denote set byUδ x⊆X U\subseteq \delta_x X , to maps an open set1U⊆X 1 U\subseteq X if tox1∈U x\in 1 U , and if to0x∈U 0 x\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 i∈Xx_i\in X, and for positive (lower) real numbers α i\alpha_i, possibly summing to one.

Borel measures

Let For more on this, seeXX$\tau$-additive measure be a topological space, and let μ\mu be a measure defined on the Borel $\sigma$-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 $\tau$-smooth.

Properties

Let XX be a topological space, and let μ\mu be a measure defined on the Borel $\sigma$-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 $\tau$-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

Let ν\nu be a valuation on a locale 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.

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 a closed subspace of XX, 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 $\tau$-additive Borel measure.

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

Theorem (see Manilla, Theorem 3.23). On every T 3T_3 topological space, a locally finite continuous valuation extends uniquely to a regular, $\tau$-additive Borel measure.

This includes in particular every metric space.

Moreover, the following is known:

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

The 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

See For also a general treatment, see

For the theory of integration over valuations, see

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 0T_0 spaces, 2004. Link here.

Revision on January 17, 2019 at 11:17:29 by Paolo Perrone See the history of this page for a list of all contributions to it.

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