A small large cardinal upper bound in $V$ for proving when certain subsets of $\omega_1$ (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that $\forall B \subseteq \omega_1 \lbrack B$ is universally Baire $\Leftrightarrow B \in L\lbrack r \rbrack$ for some real $r\rbrack$ is preserved under set-sized forcing extensions if and only if there are arbitrarily large "admissibly measurable" cardinals.