Semivariations of an additive function on a Boolean ring

Abstract

Abstract. With an additive function ϕ from a Boolean ring A into a normed space two positive functions on A, called semivariations of ϕ, are associated. We characterize those functions as submeasures with some additional properties in the general case as well as in the cases where ϕ is bounded or exhaustive.

Rosenthal L∞-theorem revisited

“…The previous variants of Rosenthal's theorem have found significant applications to Banach spaces [18,1,14] and to more general spaces ([6, Theorem 1.2], [7, p. 314 and ff. ], [5]), and to vector measures ( [12], l. drewnowski [13], [2, Lemma 4.1.37, Lemma 4.1.39, Lemma 4.1.41], [15,Proposition 3]). It is therefore natural to expect that the new, more general variants will have an even wider range of applicability.…”