Further comments on the background field method and gauge-invariant effective actions

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 (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