Hod up to AD${}_{\mathbb{R}}+\Theta$ is measurable

Abstract

Suppose $M$ is a transitive class size model of $A{D}_{\mathbb{R}}+‘‘\Theta$ is regular". $M$ is a minimal model of $A{D}_{\mathbb{R}}+‘‘\Theta$ is measurable" if (i) $\mathbb{R},Ord\subseteq M$ (ii) there is $\mu \in M$ such that $M\models ‘‘\mu$ is a normal $\mathbb{R}$-complete measure on $\Theta$" and (iii) for any transitive class size $N⊊M$ such that $\mathbb{R}\subseteq N$, $N\models ‘‘$there is no $\mathbb{R}$-complete measure on $\Theta$". Continuing Trang's work in [8], we compute HOD of a minimal model of $A{D}_{\mathbb{R}}+‘‘\Theta$ is measurable".