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

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 ⊂ $X^{s_\gamma }$ 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.