Capturing sets of ordinals by normal ultrapowers

We investigate the extent to which ultrapowers by normal measures on kappa can be correct about powersets P(lambda) for lambda > kappa. We consider two versions of this question, the capturing property CP(kappa, lambda) and the local capturing property LCP(kappa, lambda). Both of these describe the extent to which subsets of lambda appear in ultrapowers by normal measures on kappa. After examining the basic properties of these two notions, we identify the exact consistency strength of LCP(kappa, kappa+). Building on results of Cummings, who determined the exact consistency strength of CP(kappa, kappa+), and using a variant of a forcing due to Apter and Shelah, we show that CP(kappa, lambda) can hold at the least measurable cardinal.(c) 2023 Elsevier B.V. All rights reserved.