Real partitions of measure spaces
In this short note the author defines the notion of real partition and with its help existence is proved for measure-preserving mappings of several probability spaces onto the unit interval with the Lebesgue measure. A real partition of a probability space \((X, {\mathcal A}, \mu)\) is defined to be a partition \((D_t)_{t \in [0,1]}\) of the space, such that \(\mu (A^-_t) = \mu (A^+_t) = t\) for all \(t \in [0,1]\), where \(A^-_t = \cup_{s < t} D_s\) and \(A^+_t = \cup_{s \leq t} D_s\). The main theorem shows that: For every complete nonatomic probability space which includes a Cantor set, there exists a measure-preserving epimorphism of the space onto the unit interval with the Lebesgue measure.
scientific article; zbMATH DE number 794628