  We introduce a simple and straight-forward averaging procedure, which is a
generalization of one which is commonly used in electrodynamics, and show that
it possesses all the characteristics we require for linearized averaging in
general relativity and cosmology -- for weak-field and perturbed FLRW
situations. In particular we demonstrate that it yields quantities which are
approximately tensorial in these situations, and that its application to an
exact FLRW metric yields another FLRW metric, to first-order in integrals over
the local coordinates. Finally, we indicate some important limits of any
linearized averaging procedure with respect to cosmological perturbations which
are the result of averages over large amplitude small and intermediate scale
inhomogeneities, and show our averaging procedure can be approximately
implemented by that of Zotov and Stoeger in these cases.
