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A charge (finitely additive measure) defined on a Boolean algebra of setstaking values in a group $G$ is called a strictly nonzero (SNZ) charge if ittakes the identity value in $G$ only for the zero element of the Booleanalgebra. A study of such charges was initiated by Rudiger G\"obel and K.P.S.Bhaskara Rao in 2002.  Our main result is a solution to one of the questions posed in that paper: weshow that for every cardinal $\aleph$, the Boolean algebra of clopen sets of$\{0,1\}^\aleph$ has a strictly nonzero integer-valued charge. The key lemmathat we prove is that there exists a strictly nonzero integer-valuedpermutation-invariant charge on the Boolean algebra of clopen sets of$\{0,1\}^{\aleph_0}$. Our proof is based on linear-algebraic arguments, as wellas certain kinds of polynomial approximations of binomial coefficients.  We also show that there is no integer-valued SNZ charge on${\mathcal{P}}(N)$. Finally, we raise some interesting problems oninteger-valued SNZ charges.

On strictly nonzero integer-valued charges