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Erschienen in:
01.02.2010 | Original Paper
The individual ergodic theorem on the IF-events with product
Erschienen in: Soft Computing | Ausgabe 3/2010
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Abstract
The ergodic theory and particularly the individual ergodic theorem were studied in many structures. Recently the individual ergodic theorem has been proved for MV-algebras of fuzzy sets (Riečan in Czech Math J 50(125):673–680, 2000; Riečan and Neubrunn in Integral, measure, and ordering. Kluwer, Dordrecht, 1997) and even in general MV-algebras (Jurečková in Int J Theor Phys 39:757–764, 2000). The notion of almost everywhere equality of observables was introduced by Riečan and Jurečková (Int J Theor Phys 44:1587–1597, 2005). They proved that the limit of Cesaro means is an invariant observable for P-observables. In Lendelová (Int J Theor Phys 45(5):915–923, 2006c) showed that the assumption of P-observable can be omitted. In this paper we prove the individual ergodic theorem on family of IF-events and show that each
\( {\mathcal{P}} \)-preserving transformation in this family can be expressed by two corresponding \( {\mathcal{P}}^\flat,{\mathcal{P}}^\sharp \)-preserving transformations in tribe \( {\mathcal{T}}.\)