Dependent sets of a family of relations of full measure on a probability space

作者：Xiong, Jin cheng^*; Tan, Feng; Lu, Jie

来源：Science in China Series A-Mathematics, 2007, 50(4): 475-484.

DOI：10.1007/s11425-007-0025-4

• probability space
• measure-preserving transformation
• dependent set
• chaos
• dynamical system

For a probability space (X, B, mu) a subfamily F of the sigma-algebra, B is said to be a regular base if every B is an element of B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {R-gamma}(gamma is an element of Gamma) is a countable family of relations of the full measure on a probability space (X, B, mu), i.e. for every gamma is an element of Gamma there is a positive integer (s)gamma such that R gamma subset of X-s gamma with mu(s gamma) (R-gamma) = 1. In the present paper we show that if (X, B, mu) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K subset of X with mu* (K) = 1 such that (x(1),..., x(s gamma)) is an element of R-gamma for any gamma is an element of Gamma and for any s(gamma) distinct elements x(1,)....,x(s gamma) of K, where mu* is the outer measure induced by the measure mu. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.

• 出版日期2007-4
• 单位华南师范大学