Real Analysis Exchange

On Fubini-type theorems

Marek Balcerzak, Janusz Pawlikowski, and Joanna Peredko

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Abstract

We discuss some questions concerning the strengthened version of the Kuratowski-Ulam theorem obtained by Ceder. In particular, we refute Ceder’s conjecture that the measure analogue of his result holds. Further we consider mixed product \(\sigma\)-ideals \(\mathbb{K}\times \mathbb{L}\) and \(\mathbb{L}\times \mathbb{K}\) in \({\mathbb{R}}^2\) where \(\mathbb{K}\) and \(\mathbb{L}\) denote the families of meager and of Lebesgue null sets in \(\mathbb{R}\). For a set \(A\in \mathbb{K}\times \mathbb{L}\) (or \(A\in \mathbb{L} \times \mathbb{K}\)) we find large sets \(P\) and \(Q\) such that \(P\times Q\) misses \(A\). The proof is based on similar properties of \(\mathbb{K}\times\mathbb{K}\) and \(\mathbb{L}\times\mathbb{L}\) obtained by Ceder, Brodski\u i and Eggleston. A parametrized version of a Fubini-type theorem is also given.

Article information

Source
Real Anal. Exchange, Volume 21, Number 1 (1995), 340-344.

Dates
First available in Project Euclid: 3 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1341343252

Mathematical Reviews number (MathSciNet)
MR1377546

Zentralblatt MATH identifier
0857.28002

Subjects
Primary: 04A15 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]

Keywords
Fubini theorem meager set null set products of \(\sigma\)-ideal

Citation

Balcerzak, Marek; Peredko, Joanna; Pawlikowski, Janusz. On Fubini-type theorems. Real Anal. Exchange 21 (1995), no. 1, 340--344. https://projecteuclid.org/euclid.rae/1341343252


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References

  • M. Balcerzak, J. Hejduk, Density topologies for products of $\sigma$-ideals, Real Analysis Exchange, 20 (1994/95).
  • M. L. Brodski\u i, On some properties of sets of positive measure, Uspehi Mat. Nauk, 4 No. 3(31) (1949), 136–138.
  • J. Ceder, On globs, Acta Math. Hung., 43 (1984), 273–286.
  • J. Cichoń, J. Pawlikowski, On ideals of subsets of the plane and on Cohen reals, J. Symb. Logic, 51 (1986), 560–569.
  • H. G. Eggleston, Two measure properties of Cartesian product sets, Quart. J. Math. Oxford (2), 5 (1954), 108–115
  • D. H. Fremlin, The partially ordered sets of measure theory and Tukey's ordering, Note di Mathematica, 11 (1991), 177–214.
  • M. Gavalec, Iterated products of ideals of Borel sets, Colloq. Math., 50 (1985), 39–52.
  • C. G. Mendez, On sigma-ideals of sets, Proc. Amer. Math. Soc. 60 (1976), 124–128.
  • A. W. Miller, Infinite combinatorics and definability, Annals of Pure and Applied Logic, 41 (1989), 179–203.
  • J.C. Oxtoby, Measure and Category, Springer-Verlag, 1971.
  • M. Talagrand, Sommes vectorielles d'ensembles de mesure nulle, Ann. Inst. Fourier, Grenoble, 26(3) (1976), 137–172.