Eventual Capture on a Measurable Cardinal

Abstract

We continue the study from \cite{BrendleFreidmanMontoya, vandervlugtlocalizationcardinals} of localization cardinals $\mfb_κ(\in^*)$ and $\mfd_κ(\in^*)$ and their variants at regular uncountable $\hat{I}º$. We prove that if $\hat{I}º$ is measurable then these cardinals trivialize. We also provide other fundamental restrictions in the most general setting. We prove the results are optimal by forcing different values for $\mathfrak{b}_{\id^+}(\in^*),\mathfrak{d}_{\id^{++}}(\in^*)$ at a measurable. As a by-product, we prove the consistency of $\mfb_h(\in^*) < \mfb_{h'}(\in^*)$ for functions $h, h' \in \kk$, thus answering a question of Brendle, Brooke-Taylor, Friedman and Montoya. Moreover, we study the relation between these cardinals and other well-known cardinal invariants.