English translation: Real Partitions of Measure Spaces, Siberian Mathematical Journal 35 (1994), no. 1, 189-191.

Abstract:

English: The notion of real partition, introduced in the paper, presents a convenient tool for transferring many properties of the Lebesgue measure to a broader class of measure spaces. In particular, with the help of the notions introduced, we prove the existence of measure-preserving mappings from some class of probability spaces onto the unit interval with Lebesgue measure.

Informal: The key idea of this paper is the concept of a real partition: given a probability space X, a real partition of X is any partition of X into a disjoint union of measurable sets D_t indexed by real t in [0,1], such that the partial unions A^-_t and A⁺_t have measure t for every t in [0,1]. Here A^-_t is the union of D_s for all s from 0 to (but not including) t, while A⁺_t is the union of D_s for all s from 0 to t including t. Intuitively, we split the space into a continuum of disjoint annuli.

We show that every complete nonatomic probability space admits a real partition. Moreover, if there is a null set of cardinality of the continuum, then we can find a real partition (D_t) where each D_t is nonempty. Further, if now we "map" every D_t into the singleton {t}, we obtain a measure-preserving measurable epimorphism of X onto [0,1] with Lebesgue measure. (Recall that it was shown by Szpilrajn that every probability Polish space, in which all singletons have zero measure, is measure-isomorphic to [0,1] with Lebesgue measure.)

Finally, using the developed tools we present a short and simple proof that every nonatomic probability space admits a nonmeasurable set.

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