Strong Fubini properties for measure and category
Volume 178 / 2003
Abstract
Let (FP) abbreviate the statement that $$\int_0^1 \left(\int_0^1 f\, dy\right) \,dx = \int_0^1 \left(\int_0^1 f\, dx\right)\, dy $$ holds for every bounded function $f:[0,1]^2 \to {\mathbb R}$ whenever each of the integrals involved exists. We shall denote by (SFP) the statement that the equality above holds for every bounded function $f:[0,1]^2 \to {\mathbb R}$ having measurable vertical and horizontal sections. It follows from well-known results that both of (FP) and (SFP) are independent of the axioms of ZFC. We investigate the logical connections of these statements with several other strong Fubini type properties of the ideal of null sets. In particular, we establish the equivalence of (SFP) to the nonexistence of certain sets with paradoxical properties, a phenomenon that was already known for (FP). We also give the category analogues of these statements and, whenever possible, we try to put the statements in a setting of general ideals as initiated by Recław and Zakrzewski.
