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.