In the second paper under this general title, it was shown how a theorem about the torus could be deduced by a limiting process from a theorem on finite abelian groups. The object of this paper is to prove a similar continuous analogue of H. B. Mann's (α+β)-theorem. It was found that the limiting process used in the second paper could not easily be modified to apply to the present problem, and an alternative method had to be found. The method is, roughly, to prove the result first for open sets satisfying certain conditions, then for closed sets by taking intersections of open sets, and finally for arbitrary measurable sets, since every measurable set contains a closed set of almost equal measure.