UNIONS OF PRODUCTS OF INDEPENDENT SETS

We show that there exists an open set H subset of or equal to [0,1] x [0,1] with lambda(2)(H) = 1 such that for any epsilon > 0 there exists a set E satisfying lambda(1)(E) > 1/2 - epsilon and H contains the pr oduct set E x E but there is no set S with lambda(1)(S) = 1/2 and S x S subset of or equal to H. Especially this property is verified for se ts of the form H = boolean OR(i) (=) (infinity)(1) E(i) x E(i) where t he sets E(i) are independent and lambda(1)(E(i)) < 1/2. The results of this paper answer questions of M. Laczkovich and are related to a pap er of D. H. Fremlin.