1. Existence of maximin tests. Let (.¥, .91) be a Euclidean sample space, and let the distributions Pe, 8 E G, be dominated by a a-finite measure over (.¥, .91). For any mutually exclusive subsets GH , GK of G there exists a level-a test maximizing (2). [Let p = sup[infoKEecp( X»), where the supremum is taken over all level-a tests of H : 8 E GH • Let CPn be a sequence of level-a tests such that infoKEeCPn (X) tends to p. If CPn is a subsequence and cP a test (guaranteed by Theorem 3 of the Appendix) such that EeCPn(X) tends to Eecp(X) for all 8 E G, then cP is a level-a test and infoKEecp(X)'= p.)