Cyclic vectors of induced representations

In this note we prove that for a first countable locally compact group every unitary representation induced by a cyclic representation is cyclic. This result has been recently obtained also by F. Greenleaf and M. Moskowitz [2] in a more complicated way. Let G be a first countable locally compact group. Let Xtbe the space of continuous functions with compact support equipped with the Schwartz topology on G. Let D be the cone in Y' of positive-definite measures on G, i.e., ,ieD if Kx**x, ,u)>0 for all x in S. For each It in D we define I={xe3Y:Kx**x,,u)=0}. Then #?= Y-I/I is a pre-Hilbert space with a strictly positive-definite inner product (?, j),= Ky**x, ,u), where x->? is the natural mapping of MX onto >?. Moreover, if L x(h)= x(g-1h), g, heG, then I is stable under Lg, geG, and so Lg acts on 3? and is unitary with respect to (, ). As such it extends to the completion Vt of -V. Let g--L" be the representation thus obtained. If P is a projection in -' which commutes with the LI', geG, then there exists a unique measure v in D such that