Benci, Vieri; Bottazzi, Emanuele; Di Nasso, Mauro Elementary numerosity and measures. (English) Zbl 1300.26018 J. Log. Anal. 6, Paper No. 3, 14 p. (2014). Summary: In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set \(\Omega\) which takes values in a suitable non-Archimedean field and satisfies the same formal properties as finite cardinality. By improving a classic result by C. W. Henson in nonstandard analysis, we prove a general compatibility result between such elementary numerosities and measures. Cited in 7 Documents MSC: 26E30 Non-Archimedean analysis 28E15 Other connections with logic and set theory 26E35 Nonstandard analysis 60A05 Axioms; other general questions in probability Keywords:non-Archimedean mathematics; measure theory; nonstandard analysis; numerosities. PDFBibTeX XMLCite \textit{V. Benci} et al., J. Log. Anal. 6, Paper No. 3, 14 p. (2014; Zbl 1300.26018) Full Text: DOI arXiv References: [1] [1] V. Benci, I numeri e gli insiemi etichettati, Conferenza del seminario di matematica dell’Universit‘a di Bari, vol. 261 (1995), Laterza, 1–29. [2] [2] V. Benci, E. Bottazzi, M. Di Nasso, Some applications of numerosities in measure theory, in preparation. · Zbl 1309.26028 [3] [3] V. Benci, M. Di Nasso , Numerosities of labelled sets: a new way of counting, Advances in Mathematics, vol. 173 (2003), 50–67, doi:http://dx.doi.org/10.1016/ S0001-8708(02)00012-9. · Zbl 1028.03042 [4] [4] V. Benci, M. Di Nasso, M. Forti, An Aristotelian notion of size, Annals of Pure and Applied Logic, vol. 143 (2006), 43–53, doi:http://dx.doi.org/10.1016/j. apal.2006.01.008. · Zbl 1114.03055 [5] [5] A. Bernstein, F. Wattenberg, Nonstandard measure theory, in Applications of Model Theory to Algebra, Analysis and Probability (W.A.J. Luxemburg, ed.), Holt, Rinehart and Winston (1969), 171–185. [6] [6] M. Davis, Applied Nonstandard Analysis, John Wiley & Sons (1977). [7] [7] M. Di Nasso, M. Forti, Numerosities of point sets over the real line, Transactions of the American Mathematical Society, vol. 362 (2010), 5355–71, doi:http://dx.doi. org/10.1090/S0002-9947-2010-04919-0. · Zbl 1200.03033 [8] [8] R. Goldblatt, Lectures on the Hyperreals – An Introduction to Nonstandard Analysis, Graduate Texts in Mathematics, vol. 188 (1998), Springer, doi:http://dx.doi.org/ 10.1007/978-1-4612-0615-6. · Zbl 0911.03032 [9] [9] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers (6th edition, R. Heath-Brown, J. Silverman, A. Wiles, eds.), Oxford University Press (2008). · Zbl 1159.11001 [10] [10] C.W. Henson, On the nonstandard representation of measures, Transactions of the American Mathematical Society, vol. 172 (1972), 437–446, doi:http://dx.doi. org/10.1090/S0002-9947-1972-0315082-2. · Zbl 0255.28006 [11] [11] J. Yeh, Real Analysis, Theory of Measure and Integration, World Scientific Publishing Co. Pte. Ltd (2006), doi:http://dx.doi.org/10.1142/6023. · Zbl 1098.28002 [12] [12] P.A. Loeb,Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society, vol. 211 (1975), 113–122, doi:http://dx.doi.org/10.1090/ S0002-9947-1975-0390154-8. · Zbl 0312.28004 [13] [13] A. Robinson, On generalized limits and linear functionals, Pacific Journal of Mathematics, vol. 14 (1975), 269–283, doi:http://dx.doi.org/10.2140/pjm.1964. 14.269. · Zbl 0121.09502 [14] [14] D.A. Ross, Loeb measure and probability, in Nonstandard Analysis – Theory and Applications (L.O. Arkeryd, N.J. Cutland, C.W. Henson, eds.), NATO ASI Series C, vol. 493 (1997), Kluwer A.P., pp. 91–120, doi:http://dx.doi.org/10.1007/ 978-94-011-5544-1_4. · Zbl 0904.28015 [15] [15] D.A. Ross, Nonstandard measure constructions - Solutions and problems, in Nonstandard Methods and Applications in Mathematics (N.J. Cutland, M. Di Nasso, D.A. Ross, eds.), Lecture Notes in Logic, vol. 25 (2006), Association of Symbolic Logic, pp. 127–146. Universit‘a di Pisa, Italy and Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia. Dipartimento di Matematica, Universit‘a di Trento, Italy. Dipartimento di Matematica, Universit‘a di Pisa, Italy. benci@dma.unipi.it,emanuele.bottazzi@unitn.it,dinasso@dm.unipi.it http://www.dma.unipi.it/Members/benci, http://www.science.unitn.it/ bottazzi/, http://www.dm.unipi.it/ dinasso/ Received: 17 October 2013Revised: 13 August 2014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.