An uncountable Moore-Schmidt theorem

We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a "conditional" Pontryagin duality for spaces of abstract measurable maps.
Comment: 28 pages [v4]: removed material concerning weak conditional elements, following an error pointed out by the referee. Also modified the notation in line with referee comments on this paper and related papers