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In this work we extend the Fourier-Stieltjes transform of a vector measureand 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 alocally 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 ofa continuous function with compact support defined on G from the representationof G. Then, we extend the Shur orthogonality relation established for compactgroups to locally compact groups by using the representations of G induced bythe unitary representations of one of its normal compact subgroups. Thisextension enables us to develop a Fourier-Stieltjes transform in series formthat is linear, continuous, and invertible.

Fourier-Stieltjes transform defined by induced representation on locally compact groups