On strictly nonzero integer-valued charges

TL;DR

This paper constructs strictly nonzero integer-valued charges on Boolean algebras of clopen sets of infinite product spaces, solving a question from 2002, and shows their non-existence on the power set of natural numbers.

Contribution

It provides the first explicit construction of integer-valued SNZ charges on certain infinite Boolean algebras, addressing an open problem from prior research.

Findings

Existence of integer-valued SNZ charges on Boolean algebras of clopen sets of ^\u03b1.

Non-existence of such charges on ^{} (power set of natural numbers).

Use of linear algebra and polynomial approximation techniques in proofs.

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 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: 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 ${\mathcal{P}}(N)$. Finally, we raise some interesting problems on integer-valued SNZ charges.