TY - JOUR

T1 - On strictly nonzero integer-valued charges

AU - Kopparty, Swastik

AU - Bhaskara Rao, K. P.S.

N1 - Funding Information: Received by the editors August 8, 2016, and, in revised form, December 28, 2016. 2010 Mathematics Subject Classification. Primary 28B10, 03E05. The first author was supported in part by a Sloan Fellowship and NSF grants CCF-1253886 and CCF-1540634. Publisher Copyright: © 2018 American Mathematical Society.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

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U2 - https://doi.org/10.1090/proc/13700

DO - https://doi.org/10.1090/proc/13700

M3 - Article

VL - 146

SP - 3777

EP - 3789

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 9

ER -