A convexity result for the range of vector measures with applications to large economies

On a Boolean algebra we consider the topology $u$ induced by a finitely additive measure $\mu$ with values in a locally convex space and formulate a condition on $u$ that is sufficient to guarantee the convexity and weak compactness of the range of $\mu$. This result à la Lyapunov extends those obtained in (Khan, Sagara 2013) to the finitely additive setting through a more direct and less involved proof. We will then give an economical interpretation of the topology $u$ in the framework of coalitional large economies to tackle the problem of measuring the bargaining power of coalitions when the commodity space is infinite dimensional and locally convex. We will show that our condition on $u$ plays the role of the "many more agents than commodities" condition introduced by Rustichini and Yannelis in (1991). As a consequence of the convexity theorem, we will obtain two straight generalizations of Schmeidler's and Vind's Theorems on the veto power of coalitions of arbitrary economic weight.