If all subsets of cardinality less than 2"o of the real line R are Lebesgue measurable then there exists a permutation p of R with p = p-' such that on the a-field generated by S U p(s) there is no continuous probability measure. 1. Let IS I denote the cardinality of the set S. If f is a function from a set S into a set T and 1Y is a family of subsets of S then by fl(F) we denote the family {f(F): F E C}) of subsets of T. Let e be a a-field on S (i.e. a countably additive algebra of subsets of S). If X c S then e n X will denote the a-field {C n X: C E e) on X. A countably additive measure , on a countably generated a-field ( on S will be called a nontrivial continuous measure iff 0 < ,u(S) < oo and , vanishes for all atoms of e. We will say that e is measurable if there exists a nontrivial continuous measure on e. Otherwise one says that e is nonmeasurable (see [4]). Recall that a subset Y of a separable metrizable space X is called a universal null set iff for every nontrivial continuous measure , on 03x (= Borel subsets of X) we have ,u*( Y) = 0, where ,u* is the outer measure induced by ,u (see [1], [7] and [13]). It is easy to check that Y is a universal null set iff `3 y is a nonmeasurable a-field on Y (see [11] and [13]). Marczewski and Sierpin'ski discovered in [11] an uncountable universal null subset of the real line. Further information on universal null sets can be found e.g. in [1], [2], [4], [6], [7], [11] and [13], where there are also other references. A separable metric space X is called absolute Borel if X is a Borel subset of its completion. 2. We work in ZFC set theory. If X is a separable metrizable space, then we shall consider the following conditions concerning X. (i) There exists a universal null subset Y of X with I YI = IlXI. (ii) There exists a permutation p of X with p = p-1 and such that the graph of p is a universal null subset of X x X. (iii) There exists a permutation p of X with p = p-1 and such that the a-field generated by 135x U p('33x) is nonmeasurable. In connection with (i) it is worth mentioning that for X = R condition (i) is a theorem of ZFC + all subsets of R of cardinality less than 2N0 are Lebesgue Received by the editors December 15, 1978 and, in revised form, March 31, 1980. 1980 Mathematics Subject Classification. Primary 04-00, 04A15, 04A05; Secondary 28A05, 28A60, 28A10, 28A20, 28A35.