Universally Baire sets in $2^{\hat{I}}º$

Abstract

We generalize the basic theory of universally Baire sets of $2^{\ddot{I}}‰$ to a theory of universally Baire subsets of $2^{\hat{I}}º$. We show that the fundamental characterizations of the property of being universally Baire have natural generalizations that can be formulated also for subsets of $2^{\hat{I}}º$, in particular we provide four equivalent uniform definitions in the parameter $\hat{I}º$ (for $\hat{I}º$ an infinite cardinal) characterizing for each such $\hat{I}º$ the class of universally Baire subsets of $2^{\hat{I}}º$. For $\hat{I}º=\ddot{I}‰$, these definitions bring us back to the original notion of universally Baire sets of reals given by Feng, Magidor and Woodin [2].