If \(\kappa\) is measurable and GCH holds, then any ultrapower by a normal measure on \(\kappa\) will be missing some subset of \(\kappa^+\). On the other hand, Cummings showed that, starting from a \((\kappa+2)\)-strong \(\kappa\), one can force to a model (without collapsing cardinals) where \(\kappa\) carries a normal measure whose ultrapower captures the entire powerset of \(\kappa^+\). Moreover, the large cardinal hypothesis is optimal. I will present an improvement of Cummings' result and show that this capturing property can consistently hold at the least measurable cardinal.
This is joint work with Radek Honzí