  In this work we extend the Fourier-Stieltjes transform of a vector measure
and a continuous function defined on compact groups to locally compact groups.
To do so, we consider a representation L of a normal compact subgroup K of a
locally compact group G, and we use a representation of G induced by that of L.
Then, we define the Fourier-Stieltjes transform of a vector measure and that of
a continuous function with compact support defined on G from the representation
of G. Then, we extend the Shur orthogonality relation established for compact
groups to locally compact groups by using the representations of G induced by
the unitary representations of one of its normal compact subgroups. This
extension enables us to develop a Fourier-Stieltjes transform in series form
that is linear, continuous, and invertible.
