AbstractAn important statistical problem is to construct a confidence set for some functionalT(P) of some unknown probability distributionP. Typically, this involves approximating the sampling distributionJn(P) of some pivot based on a sample of sizenfromP. A bootstrap procedure is to estimateJn(P) byJn(&Pcirc;n), where P̂nis the empirical measure based on a sample of sizenfromP. Typically, one has thatJn(P) andJn(P̂n) are close in an appropriate sense. Two questions are addressed in this note. AreJn(P) andJn(P̂n) uniformly close asPvaries as well? If so, do confidence statements aboutT(P) possess a corresponding uniformity property? In the caseT(P) = P, the answer to the first questions is yes; the answer to the second is no. However, bootstrap confidence statements aboutT(P) can be made uniform over a restricted, though large, class ofP. Similar results apply to other functionalT(