MEASURE-CATEGORY PROPERTIES OF BOREL PLANE SETS AND BOREL FUNCTIONS OF TWO VARIABLES

Let I circle times J stand for the Fubini-type product of sigma-ideals I, J subset of P(R). We consider mixed measure-category sigma-ideals M circle times N and N circle times M (called the Mendez sigma-ideals), and study some features of their structure. We show that M circle times N and N circle times M are not invariant with respect to nonzero rotations. Using Fremlin's results, we describe nice Borel bases of M circle times N, N circle times M, {0} circle times N and {0} circle times M. The rest of the paper is devoted to uniform versions of the Nikodym theorem and the Lusin theorem for Borel functions of two variables.