Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## 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

## 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.

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.