On Fubini-type theorems

Abstract

We discuss some questions concerning the strengthened version of the Kuratowski-Ulam theorem obtained by Ceder. In particular, we refute Ceder’s conjecture that the measure analogue of his result holds. Further we consider mixed product \(\sigma\)-ideals \(\mathbb{K}\times \mathbb{L}\) and \(\mathbb{L}\times \mathbb{K}\) in \({\mathbb{R}}^2\) where \(\mathbb{K}\) and \(\mathbb{L}\) denote the families of meager and of Lebesgue null sets in \(\mathbb{R}\). For a set \(A\in \mathbb{K}\times \mathbb{L}\) (or \(A\in \mathbb{L} \times \mathbb{K}\)) we find large sets \(P\) and \(Q\) such that \(P\times Q\) misses \(A\). The proof is based on similar properties of \(\mathbb{K}\times\mathbb{K}\) and \(\mathbb{L}\times\mathbb{L}\) obtained by Ceder, Brodski\u i and Eggleston. A parametrized version of a Fubini-type theorem is also given.