On sigma-ideals of sets

Abstract:

Let $\Phi (\Psi )$ denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). Theorem 1. (i) $\Phi$ and $\Psi$ are $\sigma$-ideals. (ii) The union of $\Phi$ or $\Psi$ is $I \times I$. (iii) The complement of each member of $\Phi$ or $\Psi$ contains a set of power c belonging to $\Phi$ and $\Psi$, respectively, (iv) The unit square may be represented as the union of two complementary Borel sets: one in $\Phi$ and $\Psi$ and the other one of Lebesgue measure zero and first category, (v) The unit square may be represented as the union of two complementary Borel sets: one in $\Phi$ and the other one in $\Psi$. Theorem 2. $\Phi (\Psi )$ does not satisfy (vi) There is a subclass $\Upsilon$ of power $\leqslant$ c of the class $\Phi (\Psi )$ such that every member of the class is contained in some member of the subclass. Theorem 3. There does not exist a one-to-one mapping f from $I \times I$ onto itself, such that $K \in \Phi (\Psi )\;iff\;f(K)$ is a Lebesgue measure zero (first category) subset of $I \times I$. Theorems 2 and 3 hold for more general $\Phi (\Psi )$. A theorem on the theory of quotient (Boolean) algebras follows from these results.