On strictly nonzero integer-valued charges

Swastik Kopparty, K. P.S. Bhaskara Rao

Research output: Contribution to journalArticle

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} 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 languageEnglish (US)
Pages (from-to)3777-3789
Number of pages13
JournalProceedings of the American Mathematical Society
Volume146
Issue number9
DOIs
StatePublished - Jan 1 2018

    Fingerprint

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this