The strong version of the Poincaré recurrence theorem states that for any probability space , any -measure preserving transformation and any almost every point of returns to infinitely many times. In [8] (see also [4]) the theorem has been proved for MV-algebras of some type. The present paper contains a remarkable strengthening of the result stated in [8].
Independently with [7] a corresponding fuzzy approach has been developed in [3-5] with applications in measure theory. One of the results the Egoroff theorem has been proved in an abstract form. In [1] a necessary and sufficient condition for holding the Egoroff theorem was presented in the case of a space with a monotone measure. By the help of [2] and [6] we prove a variant of the Egoroff theorem stated in [4].
A probability theory on IFS-events has been constructed in [3], and axiomatically characterized in [4]. Here using a general system of axioms it is shown that any probability on IFS-events can be decomposed onto two probabilities on a Lukasiewicz tribe, hence some known results from [5], [6] can be used also for IFS-sets. As an application of the approach a variant of Central limit theorem is presented.
A variant of Alexandrov theorem is proved stating that a compact, subadditive -poset valued mapping is continuous. Then the measure extension theorem is proved for MV-algebra valued measures.
Calculus for observables in a space of functions from an abstract set to the unit interval is developed and then the individual ergodic theorem is proved.
I. Kluvánek suggested to built the Lebesgue integral on a compact interval in the real line by the help of the length of intervals only. In the paper a modification of the Kluvánek construction is presented applicable to abstract spaces, too.
Recently D. Dumitrescu ([4], [5]) introduced a new kind of entropy of dynamical systems using fuzzy partitions ([1], [6]) instead of usual partitions (see also [7], [11], [12]). In this article a representation theorem is proved expressing the entropy of the dynamical system by the entropy of a generating partition.
The paper deals with a new mathematical model for quantum mechanics based on the fuzzy set theory [1]. The indefinite integral of observables is defined and some basic properties of the integral are examined.
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