Dependent sets of a family of relations of full measure on a probability space |
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Authors: | Jin-cheng Xiong Feng Tan Jie Lü |
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Institution: | School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China |
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Abstract: | 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
⊂
with
(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, …,
) ∈ R
γ
for any γ ∈ Γ and for any s
γ
distinct elements x
1, …,
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.
This work was supported by the National Science Foundation of China (Grant No. 10471049) |
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Keywords: | probability space measure-preserving transformation dependent set chaos dynamical system |
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