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In this paper we construct a class of such partitions of the plane, including some that retain the property under various transformations. Not all known partitions are included in this class; in particular it does,not include one found by Maharam and A. H. Stone (unpublished), the first example which appears to have been considered. As side results we obtain a generalization of a theorem of Steinhaus, and a theorem on extension of a measure-preserving homeomorphism from a plane Cantor set. The property of being a member of a partition is shown to be nonexceptional in the sense of category in the space of measurable sets. We also consider the problem of partitioning a general product space. If X and Y are measure spaces, we shall say that a set E CX X Y has property (M) relative to a set UCXX Yif I,(Et)(A XB)) >0 whenever, u(A XB) >0 and A XB C U. Here u denotes the independent product measure. A partition I E, } of U will be called an (M)-partition of U if each set Ei has property (M) relative to U. Such a partition is necessarily countable if U contains a product set of positive finite measure. A subset E of a measure space with a topology is said to be metrically dense if M(En U) >0 for every nonempty open set U. It may be remarked that a measurable subset of the plane is metrically dense if and only if every equivalent set is topologically dense, and that it has property (M) relative to the plane if and only if it it metrically dense on every measurable product set that is metrically dense in itself. It is easy to construct a partition of the line or plane into a sequence of

Partitions of the plane into sets having positive measure in every non-null measurable product set