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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 lattice with a bottom element$\bottom$bottom . A element$\bottom$. A valuation or evaluation on $L$ is a map$\nu:L\to\mathbb{R}_{\ge 0}$map $\nu \colon L\to\mathbb{R}_{\ge 0}$ 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 $L$,

The real line can be replaced, depending on the context, by the space ofreal line can be replaced, depending on the context, by the space of lower reals (see for example Vickers).

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

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

Valuations on locales and topological spaces

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

Examples

Dirac valuation

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

Borel measures

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 $\tau$-smooth.

Properties

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References

See also

• Steve Vickers,A monad of valuation locales

  Steve Vickers, A monad of valuation locales (pdf).

  ,available here.

• Wikipedia, Valuation (measure theory)

• Wikipedia, Valuation (measure theory)