$AD_{\mathbb{R}}$ implies that all sets of reals are $\Theta$ universally Baire

Abstract

$AD_{\mathbb{R}}$ is the Axiom of Determinacy for games on the reals (i.e., the player play reals. We show that it implies that all sets of reals are Theta-universally Baire. As a corollary we obtain a model in which $AD$ holds and all sets of reals are fully universally Baire. Such a model was first constructed by Larson, Sargsyan and Wilson. Our result complements it as the technique is purely descriptive set theoretic.