Real Analysis Exchange

Fubini Properties of Ideals

Ireneusz Recław and Piotr Zakrzewski

Full-text: Open access

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.

Article information

Source
Real Anal. Exchange, Volume 25, Number 2 (1999), 565-578.

Dates
First available in Project Euclid: 3 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.rae/1230995393

Mathematical Reviews number (MathSciNet)
MR1778511

Zentralblatt MATH identifier
0926.03058

Subjects
Primary: Primary04A15 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] Secondary03E05

Keywords
Polish space Fubini Property Borel sets \s-ideal ccc

Citation

Recław, Ireneusz; Zakrzewski, Piotr. Fubini Properties of Ideals. Real Anal. Exchange 25 (1999), no. 2, 565--578. https://projecteuclid.org/euclid.rae/1230995393


Export citation

References

  • M. Balcerzak, Can ideals without ccc be interesting?, Topology and Appl. 55 (1994), 251–260.
  • M. Balcerzak, D. Rogowska, Making some ideals meager on sets of size of the continuum, Topology Proc. 21 (1996), 1–13.
  • T. Bartoszyński, H. Judah, Set Theory. On the structure of the real line, A. K. Peters 1995.
  • D. H. Fremlin, Real-valued-measurable cardinals, in: Set theory of the reals, Haim Judah Ed., Israel Math. Conf. Proc. 6 (1993), 151–304.
  • D. H. Fremlin and J. Jasinski, $G_\delta$-covers and large thin sets of reals, Proc. London. Math. Soc. (3) 53 (1986), 518–538.
  • M. Gavalec, Iterated products of ideals of Borel sets, Coll. Math. 50 (1985), 39–52.
  • H. Judah, A. Lior, I. Reclaw, Very small sets, Coll. Math. 72 no. 2 (1997), 207–213.
  • W. Just, M. Weese, Discovering modern set theory. II, Graduate studies in math. 18, AMS 1997.
  • A. S. Kechris, Classical descriptive set theory, Graduate Texts in Math. 156, Springer-Verlag 1995.
  • A. S. Kechris, S. Solecki, Approximating analytic by Borel sets and definable chain conditions, Israel J. Math. 89(1995), 343–356.