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

作　　者：Jin-cheng XIONG Feng TAN Jie Lü 

机构地区：[1]School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China

出　　处：《Science China Mathematics》2007年第4期475-484,共10页中国科学：数学（英文版）

基　　金：This work was supported by the National Science Fbundation of China (Grant No. 10471049)

摘　　要：For a probability space (X, B, μ) a subfamily F of the σ-algebra B is said to be a regular base if every B ∈ B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {R γ } γ∈Γ is a countable family of relations of the full measure on a probability space (X, B, μ), i.e. for every γ ∈ Γ there is a positive integer s γ such that R γ ? $X^{s_\gamma } $ with $\mu ^{s_\gamma } $ (R γ ) = 1. In the present paper we show that if (X, B, μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K ? X with μ*(K) = 1 such that (x 1, …, $x_{^{s_\gamma } } $ ) ∈ R γ for any γ ∈ Γ and for any s γ distinct elements x 1, …, $x_{^{s_\gamma } } $ of K, where μ* is the outer measure induced by the measure μ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.For a probability space (X, B,μ) a subfamily F of theσ-algebra B is said to be a regular base if every B∈B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {γr}γ∈Γis a countable family of relations of the full measure on a probability space (X,B,μ), i.e. for everyγ∈Γthere is a positive integer sγsuch that Rγ(?)Xsγwithμsγ(Rγ) = 1. In the present paper we show that if (X, B,μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K(?)X withμ*(K) = 1 such that (x1,...,xsγ)∈γr for anyγ∈Γand for any sγdistinct elements x1,..., xsγof K, whereμ* is the outer measure induced by the measureμ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.

关 键 词：probability space measure-preserving transformation dependent set chaos dynamical system 28A12 28A35 37A05 

分 类 号：O193[理学—数学]