Large cardinals with few measures

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa(+) many normal measures on the least measurable cardinal.. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of lambda strong compactness or lambda supercompactness measures on P-kappa(lambda) can be exactly lambda(+) if lambda > kappa is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.