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. ...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.展开更多
P-sets is a set pair, it is composed of internal P-set and outer P-set, it has dynamic characteristic. By using structure of P-sets, dependence theorem and identification theorem are proposed in this paper.
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 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.
基金Supported by the Natural Science Foundation of Zhumadian, Henan, China(11704)Supported by the National Natural Science Foundation of Shandong Province(60973042)Supported by the Natural Science Foundation of Shandong Province(Y2008F61, Y2008G20)
文摘P-sets is a set pair, it is composed of internal P-set and outer P-set, it has dynamic characteristic. By using structure of P-sets, dependence theorem and identification theorem are proposed in this paper.
