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. Assu...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.展开更多
基金This work was supported by the National Science Fbundation of China (Grant No. 10471049)
文摘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.
