nLab
tau-additive measure

Contents

Contents

Idea

A τ\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.

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

μ(sup λU λ)=sup λμ(U λ). \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 μ\mu on the open sets.

Examples

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

Properties

Pushforward, products and marginals

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

  • The product? of two τ\tau-additive measures on a product space can be extended to a τ\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 τ\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 μ\mu on a topological space XX, an open set UXU\subseteq X is called null for μ\mu if μ(U)=0\mu(U)=0. A set has full measure if its complement is null.

By τ\tau-additivity, the union of all null open sets is null. Its complement, which is a closed set, is called the support of μ\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

See also: Extending valuations to measures.

References

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

Last revised on October 26, 2019 at 10:24:11. See the history of this page for a list of all contributions to it.