Some Strengthenings of the Ulam Nonmeasurability Condition

The purpose of this paper is to amplify the remarks made in the footnote on p. 603 in [4]. A complete account of the investigation initiated in [4] will be published in Fundamenta Mathematicae; in this paper we shall discuss various strengthenings of the Ulam nonmeasurability condition as well as their relative strength. Some of the present results were announced in [6]. We shall assume the familiarity with notations and terminology of [4] and [5]; we shall, however, review the more frequently used terms. By an Ulam measure in a set X we shall mean a finitely additive 0-1 measure p. whose domain is a field of subsets of X containing all the one-element subsets of X and such that p({x})=0 for every xEX and p(X) = 1. A measure on X is a measure whose domain contains all subsets of X. Given a cardinal m we will denote by Xm a set of cardinality m (in a topological context Xm will denote a discrete space of cardinality m). An infinite cardinal m is said to be Ulam nonmeasurable provided that no Ulam measure on Xm is countably additive. An Ulam measure p is said to be m-additive provided that m(U$R) = sup {p(A): A EW} for every class 9t of subsets of the domain of m with card 9?^ rrt. In [4] we have considered the following conditions on an infinite cardinal m. m£M: there is a collection of sequences Af\ A2®, • • ■ ; ££E, of subsets of Xm such that card aSm and for every Ulam measure p. on Xm the equality