Extending Coarse-Grained Measures

Anna De Simone¹ ; Pavel Pták²

¹ Department of Mathematics and Statistics University Federico II, Naples Complesso Monte S. Angelo, via Cintia 80126 Naples, Italy
² Department of Mathematics Faculty of Electrical Engineering Czech Technical University 166 27 Praha 6, Czech Republic

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) no. 1, pp. 1-11

Résumé

In [4] it is proved that a measure on a finite coarse-grained space extends, as a signed measure, over the entire power algebra. In [7] this result is reproved and further improved. Both the articles [4] and [7] use the proof techniques of linear spaces (i.e. they use multiplication by real scalars). In this note we show that all the results cited above can be relatively easily obtained by the Horn–Tarski extension technique in a purely combinatorial manner. We also characterize the pure measures and settle the dimension of the normalized-measure space. We then comment on a consequence of the results for circulant matrices. Finally, we take up the case of circle coarse-grained space and also establish a measure-extension result.

Détail

DOI : 10.4064/ba54-1-1

Keywords: proved measure finite coarse grained space extends signed measure entire power algebra result reproved further improved articles proof techniques linear spaces multiplication real scalars note results cited above relatively easily obtained horn tarski extension technique purely combinatorial manner characterize pure measures settle dimension normalized measure space comment consequence results circulant matrices finally circle coarse grained space establish measure extension result