  In this paper, we will study the continuity of the Fourier transform of
measures with respect to the vague topology. We show that the Fourier transform
is vaguely discontinuous on R, but becomes continuous when restricting to a
class of Fourier transformable measures such that either the measures, or their
Fourier transforms are equi-translation bounded. We discuss continuity of the
Fourier transform in the product and norm topology. We show that vague
convergence of positive definite measures implies the equi translation
boundedness of the Fourier transforms, which explains the continuity of the
Fourier transform on the cone of positive definite measures. In the appendix,
we characterize vague precompactness of a set a measures in arbitrary LCAG, and
the necessity of second countability property of a group for defining the
autocorrelation measure.
