A -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.
-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.
Let be a topological space. A positive Borel measure on is called -additive, or -smooth, or -regular, if for every directed net of open subsets of ,
This property can be considered an instance of Scott continuity, and is equivalent to continuity of the valuation defined by on the open sets.
Every Radon measure on a Hausdorff space is -additive. This includes most regular measures of common use, such as
The pushforward measure of a -additive measure along a continuous map is itself -additive. In particular, the marginals of a -additive measure are -additive. This permits to construct a functor, even a monad (see below).
The product? of two -additive measures on a product space can be extended to a -additive measure.
One can form a measure monad analogous to the Giry monad, whose functor part assigns to a topological space the space of -additive measures over it. This monad is a submonad of the extended probabilistic powerdomain. See the measure monad on Top for more details.
Given a tau-additive measure on a topological space , an open set is called null for if . A set has full measure if its complement is null.
By -additivity, the union of all null open sets is null. Its complement, which is a closed set, is called the support of . It can be seen as the smallest closed set of full measure.
This definition can be extended to continuous valuations.
See also: Extending valuations to measures.
Last revised on October 26, 2019 at 14:24:11. See the history of this page for a list of all contributions to it.