By K. H. Kamps

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**Example text**

The existence of a bijection g : A -+ B with the graph contained in the (2 - 2)-correspondence G (defined above) implies, in the theory ZF & DC, the existence of a Lebesgue nonmeasurable function acting from R into R. ount of the result of Solovay mentioned in Remark 1, we conclude that (1) the Hall theorem cannot be proved in the theory ZF & DC; (2) Theorem 5 cannot be proved in the theory ZF & DC. We also have the next fact: 46 2 (3) it cannot be proved, in the theory ZF & DC, that there exists a linear ordering of the Vitali partition {V; : i E I}.

Clearly, Q is a subgroup of the additive group of R. Let us consider a binary relation G c R x R defined by the following formula: (x,Y)EG~X-YEQ. It is easy to see that G is an equivalence relation on the real line. The graph of this relation is a very simple subset of the Euclidean plane R 2 . Namely, it can be represented as the union of a countable family of straight lines lying in R 2 and parallel to the line {(x,y) E R2 : x = y}. Let us denote by {V; : i E l} the partition of R canonically associated with G.

Notiee that the argument presented above also proves a more general statement. In order to formulate it, we need the notion of a set of Vitali type. Let r be a subgroup of the additive group of the real line R. ally assoeiated with the equivalenee relation x ER & y ER & x - y E r. Let X be any selector of this partition. We shall say that X is a r -selector (or that X is a set of Vitali type with respect to the group r). It ean easily be seen that the preeeding argument establishes the following result.