A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF+,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF+,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF+,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF+.