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

We are interested in this theorem for two reasons. It is used in the author’s construction of generic generators (see the forthcoming [11]). Secondly the authors of [6] introduced a derived model construction producing a model of determinacy in which all sets of reals are universally Baire. The above theorem shows how to build such models from determinacy rather than large cardinals (see Corollary 5.4). Along the way of proving Theorem 0.1, we fill in some gaps in descriptive inner model theory. In particular, we outline the process of getting N ∗ x -like models used in [12] (see [12, Theorem 2.25]). We still leave one gap. The proof of Theorem 1.7 is not given as the full proof will go beyond the scope of this paper. However, we believe we have given enough outline so that the interested reader can prove it. I would like to thank the anonymous referee for a long list of corrections. The author was supported by the NSF Career Award DMS-1352034.