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 {Rγ}γ∈r is a countable family of relations of the full measure on a probability space (X, (B), μ), i.e. for every γ ∈ Г there is a positive integer sγ such that Rγ (∩) Xsγ with μsγ(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 with μ*(K) = 1 such that (x1,...,xsγ) ∈Rγ for any γ ∈ Г and for any sγ distinct elements x1,... ,xsγ 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.