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

TL;DR

The paper proves that under the Axiom of Determinacy for reals, all sets of reals are Theta-universally Baire, providing a descriptive set theoretic approach to a known model where AD holds and all reals are universally Baire.

Contribution

It demonstrates that $AD_{ eals}$ implies all sets of reals are Theta-universally Baire using purely descriptive set theoretic methods, complementing existing models.

Findings

All sets of reals are Theta-universally Baire under $AD_{ eals}$.

Constructs a model where $AD$ holds and all reals are universally Baire.

Provides a descriptive set theoretic proof technique.

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.