On strictly nonzero integer-valued charges

Abstract:

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 Rüdiger Göbel and K. P. S. Bhaskara Rao in 2002.

Our main result is a solution to one of the questions posed in that paper: we show that for every cardinal $\aleph$, the Boolean algebra of clopen sets of $\{0,1\}^\aleph$ 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 $\{0,1\}^{\aleph _0}$. Our proof is based on linear-algebraic arguments, as well as certain kinds of polynomial approximations of binomial coefficients.

We also show that there is no integer-valued SNZ charge on ${\mathcal {P}}(\mathbb {N})$. Finally, we raise some interesting problems on integer-valued SNZ charges.