Capturing sets of ordinals by normal ultrapowers

We investigate the extent to which ultrapowers by normal measures on $\kappa$ can be correct about powersets $\mathcal{P}(\lambda )$ for $\lambda >\kappa$. We consider two versions of this questions, the capturing property $\mathrm{C}\mathrm{P}(\kappa ,\lambda )$ and the local capturing property $\mathrm{L}\mathrm{C}\mathrm{P}(\kappa ,\lambda )$. $\mathrm{C}\mathrm{P}(\kappa ,\lambda )$ holds if there is an ultrapower by a normal measure on $\kappa$ which correctly computes $\mathcal{P}(\lambda )$. $\mathrm{L}\mathrm{C}\mathrm{P}(\kappa ,\lambda )$ is a weakening of $\mathrm{C}\mathrm{P}(\kappa ,\lambda )$ which holds if every subset of $\lambda$ is contained in some ultrapower by a normal measure on $\kappa$. After examining the basic properties of these two notions, we identify the exact consistency strength of $\mathrm{L}\mathrm{C}\mathrm{P}(\kappa ,{\kappa }^{+})$. Building on results of Cummings, who determined the exact consistency strength of $\mathrm{C}\mathrm{P}(\kappa ,{\kappa }^{+})$, and using a forcing due to Apter and Shelah, we show that $\mathrm{C}\mathrm{P}(\kappa ,\lambda )$ can hold at the least measurable cardinal.