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. In this paper we investigate the following problem: Suppose G is a locally compact group' and let N be a closed normal subgroup of G. What hypotheses are needed on G and N in order that each element W of the dual space G of G should be weakly contained in some induced representation of the form Ux, where x is an element of the dual space N of N? This is a generalization of the classical question: What conditions must we put on a locally compact group G in order that each of its irreducible unitary representations should be weakly contained in the regular representation of G? What we show here is that the generalized question may be reduced, in the presence of some quite general assumptions, to the classical one. Our presentation leans heavily on the analysis of G in terms of ? 'a la Mackey, and we therefore do not claim the simplest possible proof. However, consideration of this question seems to indicate a necessity for using some kind of detailed analysis of representations. In ?1 we give a sketch of the particular version of Mackey's theory which we will use. We are then able to state the "quite general assumptions" mentioned above. In ?2 we prove the equivalence of two representations. This equivalence is not needed in the proofs of the main theorems of this paper, but it is of interest in itself and will be referred to in a subsequent paper. In ?3 the idea of "weak containment" for cocycle representations is presented, and in ?4 we give the main theorems.

A weak containment theorem for groups with a quotient 𝑅-group