On Strictly Nonzero Integer-valued Charges

A charge (finitely additive measure) defined on a Boolean algebra of sets taking values in a group G is called a strictly nonzero (SNZ) charge if it takes the identity value in G only for the zero element of the Boolean algebra. A study of such charges was initiated by Rudiger Göbel and KPS Bhaskara Rao in 2002 [3]. Our main result is a solution to one of the questions posed in that paper: we show that for every cardinal א, the Boolean algebra of clopen sets of {0, 1}א has a strictly nonzero integer-valued charge. The key lemma that we prove is that there exists a strictly nonzero integer-valued permutation-invariant charge on the Boolean algebra of clopen sets of {0, 1}א0 . Our proof is based on linearalgebraic arguments, as well as certain kinds of polynomial approximations of binomial coefficients. We also show that there is no integer-valued SNZ charge on P(N). Finally, we raise some interesting problems on integer-valued SNZ charges.