  The aim of this paper is to give a firm and clear proof of the existence in
the background field framework of a gauge invariant effective action for any
gauge theory ({\it background gauge equivalence}). Here by effective action we
mean a functional whose Legendre transform restricted to the physical shell
generates the matrix elements of the connected $S$-matrix. We resume and
clarify a former argument due to Abbott, Grisaru and Schaefer based on the
gauge-artifact nature of the background fields and on the identification of the
gauge invariant effective action with the generator of the proper, background
field, vertices.
