nLab valuation (measure theory)

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

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Idea

A valuation is a construction analogous to that of a measure, which is defined on a topological space rather than on a measurable space. As such, it is more compatible with constructive mathematics, and readily generalizable to contexts such as point-free topology.

On most spaces of interest for measure theory and probability (such as metric spaces) the notions of suitably continuous valuations and measures coincide (see below).

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 $L$,

$$\nu(x) + \nu(y) = \nu(x \vee y) + \nu(x \wedge y) .$$

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

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

$$\nu \big( \sup_{\lambda} x_\lambda \big) = \sup_\lambda \nu(x_\lambda) .$$

Note that, differently from measures, there is no explicit mention of complements. Moreover, the continuity condition can be interpreted, measure-theoretically, both as an analogue of $\sigma$-additivity, and as an analogue of a regularity condition (more precisely it corresponds to τ-additivity). See correspondence between measure and valuation theory for more on this.

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

$$\delta_x(U) \;\coloneqq\; \begin{cases} 1 & x\in U \\ 0 & x\notin U. \end{cases}$$

These are the valuation analogue of Dirac measures, and give the unit map of valuation monads. Note the analogy with neighborhood filters: Dirac measures can be considered a quantitative analogue thereof.

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

$$\nu = \sum_{i=1}^n \alpha_i\,\delta_{x_i} ,$$

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

Valuations share a number of constructions which are similar to those for measures. The constructions below are given for the case of continuous valuations on topological spaces.

Pushforward

Given spaces $X$ and $Y$ and a continuous map $f:X\to Y$, we can map a valuation $\nu$ on $X$ to a valuation $f_*\nu$ on $Y$ in the following way. For every open set $U\subseteq Y$, we set

$$f_*\nu (U) \coloneqq \nu(f^{-1}(U)) .$$

The assignment gives a well-defined valuation, which is continuous if $\nu$ is continuous. We call $f_*\nu$ the pushforward valuation of $\nu$ along $f$.

Compare with the analogous construction for measures.

This construction allows to makes the assignment functor, which is even a monad (see measure monads for more on this).

Joints, marginals and products

Given a valuation $\nu$ on a product space $X\times Y$, one calls its pushforward along the projection map $X\times Y\to X$ the marginal of $\nu$ on $X$. The same can be done for $Y$. The assignment from $\nu$ to the ordered pair of its marginals can be seen as an oplax monoidal structure of the valuation monad.

Given valuations $\nu$ on $X$ and $\rho$ on $Y$, there may be many possible valuations on $X\times Y$ which have $\nu$ and $\rho$ as their marginals. Any such valuation on $X\times Y$ is called a joint valuation or coupling of $\nu$ and $\rho$.

Between the many joints of $\nu$ and $\rho$ there is always a canonical choice, namely the product valuation $\nu\otimes\rho$. From the point of view of probability theory, this corresponds to a distribution exhibiting independence? between $\nu$ and $\rho$. On the basis of the product topology of $X\times Y$ given by sets in the form $U\times V$ for $U\subseteq X$ and $V\subseteq Y$ open, the product valuation is given by

$$\nu\otimes\rho (U\times V) \coloneqq \nu(U)\cdot\rho(V).$$

Note that this is not enough to define a valuation a priori. For continuous valuations, however, this turns out to be the well-defined, as proven for example in Heckmann ‘96.

The assignment to $\nu$ and $\rho$ of their product valuation can be seen as a lax monoidal structure of the valuation monad. See also monads of probability, measures and valuations#monoidal_structure.

Null sets and support

Valuations admit a notion of support similar 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 $\nu$.

The support induces a morphism of monads (see Fritz-Perrone-Rezagholi, section 5).

Integration

There is an integration theory for valuation analogous to that of measures, where the open sets play the role of the measurable sets, and lower semicontinuous functions play the role of measurable functions (see also correspondence between measure and valuation theory). Sometimes integration of valuation is known as lower integration, since approximations are done from below.

The way to define integration, mutatis mutandis, parallels usual Lebesgue integral construction. We sketch the construction for the case of topological spaces.

Let $\nu$ be a valuation on a space $X$.

• Given an open set $U\subseteq X$, and denoting by $1_U$ its indicator function, we define

  $$\int 1_U \, d\nu \;\coloneqq\; \nu(U) .$$

• A simple lower semicontinuous function is a lower semicontinuous function assuming only finitely many values. Such functions can be expressed (nonuniquely) as finite positive linear combinations of indicator functions:

  $$f \;=\; \sum_i r_i \, 1_{U_i} .$$

  We define the integral of a simple $f$ as

  $$\int f \, d\nu \;\coloneqq\; \sum_{i}r_i \nu(U_i) .$$

  This is well-defined, i.e.~it depends only on $f$ and not on the particular way of expressing $f$ as a linear combination of indicators.

• Every lower semicontinuous function can be written as pointwise directed supremum of simple lower semicontinuous functions. So suppose $g:X\to[0,\infty]$ is lower semicontinuous. Take an increasing net $(g_\alpha)_{\alpha\in A}$ of nonnegative simple lower semicontinuous functions. Then we define

  $$\int g \, d\nu \;\coloneqq\; \sup_\alpha \int g_\alpha\,d\nu ,$$

  where the integral on the right is the one defined above for simple functions, and the supremum on the right is either the one of real numbers (or lower real numbers), or $+\infty$.

This integral satisfies analogous properties to the Lebesgue integral, such as linearity and Scott continuity (cfr. the sequential monotone continuity? of the Lebesgue integral).

Monads of valuations

Just as there are several monads of measures (such as the Giry monad), there are a number of analogous monads of valuations. The most famous are

• The extended probabilistic powerdomain on the category of topological spaces, which was introduced by Heckmann 96.

• The valuation monad on locales?, defined by Steve Vickers.

• The probabilistic powerdomain? on the category of dcpos, defined by Jones and Plotkin, of wide use in theoretical computer science.

See also the list at monads of probability, measures, and valuations.

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

A locally finite continuous valuation on a regular topological space extends uniquely to a regular τ-smooth Borel measure. A locally finite continuous valuation on a locally compact sober space extends uniquely to a τ-smooth Borel measure.

(Manilla 02, Theorems 4.4 and 4.12)

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.

Particular cases

• Every Dirac valuation can be extended to the corresponding Dirac measure.

• The pushforward of an extendable valuation along a continuous map is again extendable, and the resulting measure is the pushforward measure of the extension. (Equivalently, the restriction of measures to valuations is a natural transformation).

• The above specializes to the fact that marginals of extendable valuations are extendable, and the resulting measure is the marginal measure.

• Somewhat conversely, the product of extendable valuations is extendable, and the resulting measure is the product measure? of the extensions.

• The integration, given by the multiplication map $E: P P X \to P X$ of the extended probabilistic powerdomain, maps extendable valuations to extendable valuations, and the resulting measure is the integral of the extension. Therefore there is a morphism of monads exhibiting τ-additive measures as a submonad of valuations. (See also the measure monad on Top.)

See also

• monads of probability, measures, and valuations
• correspondence between measure and valuation theory
• monad, Giry monad, monads in computer science
• extended probabilistic powerdomain, probabilistic powerdomain?, valuation monad on locales?
• probability valuation
• sigma-continuous valuation, sigma-continuous probability valuation
• content (measure theory), probability content?
• measure, τ-additive measure, probability measure

References

• Reinhold Heckmann, Spaces of valuations, Annals of the New York Academy of Sciences, 806 1 Papers on General Topology and Applications, (1996) $[$doi:10.1111/j.1749-6632.1996.tb49168.x, pdf$]$

• Mauricio Alvarez-Manilla, Achin Jung, Klaus Keimel, The probabilistic powerdomain for stably compact spaces, Theoretical Computer Science 328, 2004 (doi:10.1016/j.tcs.2004.06.021)

• Jean Goubault-Larrecq and Xiaodong Jia, Algebras of the extended probabilistic powerdomain monad, ENTCS 345, 2019

  (doi:10.1016/j.entcs.2019.07.015)

• Tobias Fritz, Paolo Perrone and Sharwin Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, 2019 (arXiv:1910.03752)

For valuations on locales, see

• Steve Vickers, A monad of valuation locales, 2011 (pdf)

For the theory of integration over valuations, see

• Olaf Kirch, Bereiche und Bewertungen (in German), Master Thesis, Technische Hochschule Darmstadt, 1993 (ps.gz)

• Achim Jung, Stably compact spaces and the probabilistic powerspace construction, ENTCS 87, 2004 (doi:10.1016/j.entcs.2004.10.001).

• Thierry Coquand and Bas Spitters, Integrals and Valuations, 2009, Logic and Analysis (2009) 1(3) p.1-22 (arXiv:0808.1522)

Integration on locales can be found in

• Steve Vickers, A localic theory of lower and upper integrals, 2008 (doi:10.1002/malq.200710028)

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 (doi:10.1016/S1571-0661(05)80210-5)

• Mauricio Alvarez-Manilla, Measure theoretic results for continuous valuations on partially ordered spaces, Dissertation, 2000 (ps.gz)

• Mauricio Alvarez-Manilla, Extension of valuations on locally compact sober spaces, Topology and its Applications Volume 124, Issue 3, 20 October 2002, Pages 397-433 (doi:10.1016/S0166-8641(01)00249-8)

• Alex Simpson, Measure, randomness and sublocales.

• Klaus Keimel and Jimmie D. Lawson?, Measure extension theorems for

  $T_0$ spaces, 2004 (doi:10.1016/j.topol.2004.02.019)