### 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 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 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 P(N). Finally, we raise some interesting problems on integer-valued SNZ charges.

Original language | English (US) |
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Pages (from-to) | 3777-3789 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 9 |

DOIs | |

State | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*146*(9), 3777-3789. https://doi.org/10.1090/proc/13700