Measure-category properties of Borel plane sets and Borel functions of two variables

Let [graphic not available: see fulltext] stand for the Fubini-type product of σ-ideals [graphic not available: see fulltext]. We consider mixed measure-category σ-ideals [graphic not available: see fulltext] and [graphic not available: see fulltext] (called the Mendez σ-ideals), and study some features of their structure. We show that [graphic not available: see fulltext] and [graphic not available: see fulltext] are not invariant with respect to nonzero rotations. Using Fremlin’s results, we describe nice Borel bases of [graphic not available: see fulltext] and [graphic not available: see fulltext]. The rest of the paper is devoted to uniform versions of the Nikodym theorem and the Lusin theorem for Borel functions of two variables.