The aim of this paper is the investigation of the topological properties of the range of a finitely additive measure with values in a topological group. Via an example the standard strong continuity (or s-bundedness) versus the assumption of semiconvexity are investigated, and since they prove to be not comparable in the framework of topological groupsc(contrary to the case of Banach valued, and more in general LCTVS-valued finitely additive measures) results concerning arcwise connectedness and relative compactness of the range are given under both assumptions, by means of ad hoc extension techniques. The result in the semiconvex case is proven along an ad hoc constructive technique.
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