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An archimedean -group is called epicomplete (or universally σ-complete, or sequentially inextensible) if it is divisible, σ-complete and laterally σ-complete.Various characterizations of such G are known in case the G have weak order units. The "theorem" of the title is a characterization of such G which have no weak order unit; it involves the requirement that G have a certain kind of representation. The "question" of the title is whether every epicomplete G has such a representation.

In a category C an object G is epicomplete if the only epic monics out of G are isomorphisms, epic or monic meant in the categorical sense of right or left cancellable. For each of the categories Arch: archimedean -groups with -homomorphisms, and its companion category W: Arch-objects with distinguished weak unit and unit-preserving -homomorphisms, (and for the corresponding categories of vector lattices) epicompleteness has been characterized as divisible and conditionally and laterally σ-complete, and it has been shown to be monoreflective.Denote the reflecting functors by β and β W , respectively. What are they? For W the Yosida representation has been used to realize β W A as a certain quotient of B (Y A), the Baire functions on the Yosida space of A. For Arch, very little has been known. Here we give a general representation theorem, Theorem A, for βG as a certain subdirect product of W -epicomplete objects derived from G. That result, some W -theory, and the relation between epicity and relative uniform density are then employed to show Theoremis the -group of continuous functions on Y with compact support and B L (Y ) is the -group of Baire functions on Y having Lindelöf cozero sets.