Fubini Properties of Ideals

Abstract

Let $I$ and $J$ be \s-ideals on Polish spaces $X$ and $Y$, respectively. We say that the pair $\langle I,J\rangle$ has the Fubini Property (FP) if for every Borel subset $B$ of $X\times Y$, if all its sections $B_x= \{y\: \langle x,y\rangle\in B\}$ are in $J$, then its sections $B^y=\{x\: \langle x,y\rangle\in B\}$ are in $I$, for every $y$ outside a set from $J$. We study the question, which pairs of $\sigma$-ideals have the Fubini Property. We show, in particular, that:

-- $\langle$ MGR$(X), J\rangle$ satisfies FP, for every $J$ generated by any family of closed subsets of $Y$ (MGR$(X)$ is the $\sigma$-ideal of all meager subsets of $X$),

-- $\langle$ NULL$_\mu, J \rangle$ satisfies FP, whenever $J$ is generated by any of the following families of closed subsets of $Y$ (NULL$_mu$ is the $\sigma$-ideal of all subsets of $X$, having outer measure zero with respect to a Borel $\sigma$-finite continuous measure $\mu$ on $X$):

(i) all closed sets of cardinality $\leq 1$,

(ii) all compact sets,

(iii) all closed sets in NULL$_\nu$ for a Borel \s-finite continuous measure $\nu$ on $Y$,

(iv) all closed subsets of a $\hbox { $\mathbf{\Pi}^1_1$}$ set $A\subseteq Y$.

We also prove that $\langle$MGR$(X)$, MGR$(Y)\rangle$ and $\langle$ NULL$_\mu$, NULL$_\nu\rangle$ are essentially the only cases of FP in the class of \s-ideals obtained from MGR$(X)$ and NULL$_\mu$ by the operations of Borel isomorphism, product, extension and countable intersection.