Positive Definite Measures

In this paper we prove two theorems relating positive definite measures to induced representations. The first shows how the injection of a positive definite measure on a topological group H into a containing locally compact group G in which H is closed gives rise to induced representations. The second is another version of Mackey's imprimitivity theorem, along the lines of Loomis' proof [5]. We feel this is justified on several grounds. Firstly, our proof is simpler than Loomis'. We make no use of the Radon-Nikodym theorem nor of quasi-invariant measures. Secondly, we do not assume in advance that our system of imprimitivity is based on the reduced algebra of Borel sets in G/H. Instead, this fact is seen as a consequence of the theorem. Finally, the statement and proof of Theorem 2 in [5] are in need of minor repairs. Using Loomis' notation, the induced representation space of Fis spanned, not by the set of functions {fu-uEH}, but rather by the set {[£]/„: uEH, E a Borel subset of G/K}. Formula (8) must then be replaced by formula (11) in the statement of the theorem. The algebra Co(SXG) used in the present paper may be looked upon as a device for accomplishing these changes. All nonobvious definitions, notations, and conventions are those of [1].