Borel Globalizations of Partial Actions of Polish Groups

TL;DR

This paper proves that the enveloping space of a partial Polish group action is a standard Borel space, extending known results about group actions and Borel selectors to partial actions.

Contribution

It generalizes Burgess's theorem on Borel selectors and extends properties of Vaught's transform to partial group actions, establishing the Borel structure of the enveloping space.

Findings

The enveloping space of a partial Polish group action is a standard Borel space.

Generalization of Burgess's theorem for Borel selectors to partial actions.

Properties of Vaught's transform are valid for partial group actions.

Abstract

We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $\tau$ on $X_G$ such that $(X_G, \tau)$ is Polish and the quotient Borel structure on $X_G$ is equal to $Borel(X_G,\tau)$. To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups.