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ON STRICTLY NONZERO INTEGER-VALUED CHARGES.
- Source :
-
Proceedings of the American Mathematical Society . Sep2018, Vol. 146 Issue 9, p3777-3789. 13p. - Publication Year :
- 2018
-
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 Ρ(ℕ). Finally, we raise some interesting problems on integer-valued SNZ charges. [ABSTRACT FROM AUTHOR]
- Subjects :
- *INTEGERS
*BOOLEAN algebra
*POLYNOMIALS
*BINOMIAL coefficients
*REAL numbers
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 146
- Issue :
- 9
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 130701256
- Full Text :
- https://doi.org/10.1090/proc/13700