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₊.