Borel Globalizations of Partial Actions of Polish Groups

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We show that the enveloping space XG 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 τ on XG such that (XG, τ) is Polish and the quotient Borel structure on XG is equal to Borel(XG,τ). 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.