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

The real 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 $\{x_\lambda\}_{\lambda\in\Lambda}$ 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 $\mathcal{O}(L)$. In particular, a valuation on a 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 (pdf).

• Wikipedia, Valuation (measure theory)