nLab tau-additive measure

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Idea

A $\tau$-additive measure (alias τ-regular, τ-smooth) is a measure on a topological space which interacts well with the topology. In particular, it has a well-defined and well-behaved support.

$\tau$-additive measures can be thought of as a point of contact between measure theory, and the theory of valuations. More on this in correspondence between measure and valuation theory.

Definition

Let $X$ be a topological space. A positive Borel measure $\mu$ on $X$ is called $\tau$-additive, or $\tau$-smooth, or $\tau$-regular, if for every directed net $\{U_\lambda\}_{\lambda\in\Lambda}$ of open subsets of $X$,

$$\mu \big( \sup_{\lambda} U_\lambda \big) = \sup_\lambda \mu(U_\lambda) .$$

This property can be considered an instance of Scott continuity, and is equivalent to continuity of the valuation defined by $\mu$ on the open sets.

Examples

Every Radon measure on a Hausdorff space is $\tau$-additive. This includes most regular measures of common use, such as

• the Dirac measures,
• the Lebesgue measure on the real line,
• the measure induced by the volume form of a Riemannian manifold,
• the Haar measure on a Lie group (or more generally a locally compact Hausdorff topological group).

Properties

Pushforward, products and marginals

• The pushforward measure of a $\tau$-additive measure along a continuous map is itself $\tau$-additive. In particular, the marginals of a $\tau$-additive measure are $\tau$-additive. This permits to construct a functor, even a monad (see below).

• The product? of two $\tau$-additive measures on a product space can be extended to a $\tau$-additive measure.

Monad structure

One can form a measure monad analogous to the Giry monad, whose functor part assigns to a topological space the space of $\tau$-additive measures over it. This monad is a submonad of the extended probabilistic powerdomain. See the measure monad on Top for more details.

Null sets and support

Given a tau-additive measure $\mu$ on a topological space $X$, an open set $U\subseteq X$ is called null for $\mu$ if $\mu(U)=0$. A set has full measure if its complement is null.

By $\tau$-additivity, the union of all null open sets is null. Its complement, which is a closed set, is called the support of $\mu$. It can be seen as the smallest closed set of full measure.

This definition can be extended to continuous valuations.

Relationship with other measure-theoretical notions

• Every Radon measure on a Hausdorff space is $\tau$-additive.
• Every $\tau$-additive measure on a compact Hausdorff space is Radon.
• Every $\tau$-additive measure can be restricted to a continuous valuation.
• Every continuous valuation on a regular Hausdorff space or on a locally compact sober space can be extended to a $\tau$-additive measure.

See also: Extending valuations to measures.

Related notions

• Borel measure

• Radon measure

• continuous valuation

• extended probabilistic powerdomain

• correspondence between measure and valuation theory

References

• V. Bogachev, Measure Theory, vol. 2 (2007).