Thomas E. Armstrong

Arrow's theorem with restricted coalition algebras☆

Abstract Arrows Theorem is shown to be valid for measurable spaces of economic agents even with only a Boolean algebra of measurable coalitions. Purely competitive social welfare functions which ignore negligible coalitions may be handled by passing to a Boolean quotient algebra with the ideal of negligible coalitions. Invisible dictators are ‘agents’ in the Stone space of the quotient algebra, in which context social welfare functions induce continuous preference profiles.

When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

It is shown that the algebra of regular sets for a finitely additive Borel measure p. on a compact Hausdorff space is a a-algebra only if it includes the Baire algebra and p is countably additive on the o-algebra of regular sets. Any infinite compact Hausdorff space admits a finitely additive Borel measure whose algebra of regular sets is not a o-algebra. Although a finitely additive measure with a a-algebra of regular sets is countably additive on the Baire o-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a o-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, l}-valued countably additive measure from a o-algebra to a larger a-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex. 1980 Mathematics subject classification (Amer. Math. Soc): 28 C 15, 28 A 60, 54 G 10.

Erratum to "Liapounoff's Theorem for Nonatomic, Bounded, Finitely-Additive, Finite-Dimensional, Vector-Valued Measures"

It is erroneously stated, in the note added in proof on p. 514, that the support of a Radon measure on a quasi-F-space is Stonian. Frederick K. Dashiell gives a counterexample, Example 3.8 on p. 412 of his paper Nonweakly compact operators from order-Cauchy complete C(S) lattices, with application to Baire classes, Trans. Amer. Math. Soc. 266 (1981), 397-413. This counterexample measure is in fact nonatomic. It is not known whether Liapounoffs convexity theorem is valid for quasi-F-algebras. It is not known whether it is necessary for the validity of Liapounoffs convexity theorem on a Boolean algebra J1 that every nonatomic Radon measure on the Stone space Xs must have Stonian support. A characterization of those compact Hausdorff spaces X (or just the totally disconnected ones) so that every nonatomic measure has Stonian support is not known.