• A. W.~Apter, J. Cummings, and J. D. Hamkins, “Large cardinals with few measures,” Proc.~Amer.~Math.~Soc., vol. 135, iss. 7, p. 2291–2300, 2007.
  [Bibtex]

  @ARTICLE{ApterCummingsHamkins2006:LargeCardinalsWithFewMeasures,
  AUTHOR = {Arthur W.~Apter and James Cummings and Joel David Hamkins},
  TITLE = {Large cardinals with few measures},
  JOURNAL = {Proc.~Amer.~Math.~Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
  VOLUME = {135},
  YEAR = {2007},
  NUMBER = {7},
  PAGES = {2291--2300},
  ISSN = {0002-9939},
  CODEN = {PAMYAR},
  MRCLASS = {03E35 (03E55)},
  MRNUMBER = {2299507 (2008b:03067)},
  MRREVIEWER = {Tetsuya Ishiu},
  DOI = {10.1090/S0002-9939-07-08786-2},
  URL = {http://jdh.hamkins.org/largecardinalswithfewmeasures/},
  eprint = {math/0603260},
  archivePrefix = {arXiv},
  primaryClass = {math.LO},
  file = F,
  }

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 $\kappa$. 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.