Strong Fubini properties for measure and category

Krzysztof Ciesielski¹ ; Miklós Laczkovich²

¹ Department of Mathematics West Virginia University Morgantown, WV 26506-6310, U.S.A.
² Department of Analysis Eötvös Loránd University Pázmány Péter sétány 1//C 1117 Budapest, Hungary and Department of Mathematics University College London WC1E 6BT London, England

Fundamenta Mathematicae, Tome 178 (2003) no. 2, pp. 171-188

Résumé

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.

Détail

DOI : 10.4064/fm178-2-6

Keywords: abbreviate statement int int right int int right holds every bounded function mathbb whenever each integrals involved exists shall denote sfp statement equality above holds every bounded function mathbb having measurable vertical horizontal sections follows well known results sfp independent axioms zfc investigate logical connections these statements several other strong fubini type properties ideal null sets particular establish equivalence sfp nonexistence certain sets paradoxical properties phenomenon already known category analogues these statements whenever possible try put statements setting general ideals initiated rec zakrzewski