SIGMA-POROUS SETS IN PRODUCTS OF METRIC SPACES AND SIGMA-DIRECTIONALLY POROUS SETS IN BANACH SPACES

Abstract

We show that no reasonable classical form of ``Fubini type theorems'' can hold for the $\sigma$-ideal of $\sigma$-porous sets in products of metric spaces (even in the plane). Then we prove that a ``Fubini type theorem'' in a weak decomposition form remains true also for this $\sigma$-ideal, and we illustrate how this fact may be applied to the study of the behavior of measures on small sets in product spaces. We also prove an analogical decomposition theorem for $\sigma$-directionally porous sets in Banach spaces; such sets arise naturally as exceptional sets in some questions concerning differentiability properties of Lipschitz functions on Banach spaces.