  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.
