enemy. Some of the theorems in the first two sections are modeled on those of Snapper, in [18], who considered these problems for everything discrete, the powerful techniques of homological algebra being available in this case. Unfortunately, there is no machinery of projective or injective resolutions immediately at our disposal for the cohomology theories based on Borel cochains. There is a type of resolution, which leans toward the injective, but its full homological character is not understood as yet. The entire idea of considering the cohomology of general group actions was inspired by the paper of Snapper. These things have been considered in the continuous case by Mostow in [17], where equivariant cohomology groups are defined and studied. Almost nothing being known about the higher cohomology groups, it was desirable to ferret out those isolated pockets of triviality. This is done in the last section, where it is shown that for G abelian locally compact separable metrizable, R the real line, G operating trivially on R, Hn(G, R), the bounded Borel groups defined in [16], do not depend upon the topology of G, nor its torsion. In fact, R seems to ignore everything about G except a certain torsion-free discrete group D associated to G. The further fact that these groups are isomorphic to the group of alternating multilinear functions An (G, R) was suggested by the results of Kleppner on H2(G, T), where G is locally compact abelian and T the circle group. He showed that for all G except a certain intractable class of 2-primary groups, whose fate is yet undecided, that H2(G, T)=L2(G, T)/S2(G, T), the continuous bihomomorphisms modulo the symmetric ones. We have not resolved this open question,