Universally Baire Sets of Reals

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Abstract

We introduce a generalization of the Baire property for sets of reals via the notion that a set of reals is universally Baire. We show that the universally Baire sets can be characterized in terms of their possible Souslin representations and that in the presence of large cardinals every universally Baire set is determined. We also study the connections between large cardinals, generalizations of \( {\underset{\raise0.3em\hbox{\(\smash{\scriptscriptstyle\thicksim}\)}}{\sum }}_{2}^{1} \) absoluteness with respect to set generic extensions, and various sets being universally Baire.