Abstract
In the unit square [0, 1] ×[0, 1] endowed with the Lebesgue measure λ, we construct a Borel subset A with the following property: if U and V are any two non-negligible Borel subsets of [0, 1], then \(0 <\lambda {\bigl ( A \cap (U \times V )\bigr )} <\lambda (U \times V )\).
The support of the ANR programme ProbaGeo is gratefully acknowledged.
[…] tout événement est comme un moule d’une forme particulière […]. M. Proust, Albertine disparue.
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- 1.
All linear notions (dimension, rank, independence) are understood with \(\mathbb{F}\) as the field of scalars.
- 2.
A weaker hypothesis suffices, namely, for each i ∈ r there is at least one h ∈ H with \(h \cap r =\{ i\}\).
References
C. Dellacherie, M. Émery, Filtrations indexed by ordinals; application to a conjecture of S. Laurent. Séminaire de Probabilités XLV, Springer Lecture Notes in Mathematics, vol. 2078 (Springer, Berlin, 2013), pp. 141–157
J. Neveu, Mathematical Foundations of the Calculus of Probability (Holden-Day, San Francisco, 1965)
Acknowledgements
I thank Claude Dellacherie for many useful comments, Gaël Ceillier, Jacques Franchi, Nicolas Juillet and Vincent Vigon for a sizeable simplification of the proof, and Wilfrid S. Kendall who suggested the term “coded”.
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© 2013 Springer International Publishing Switzerland
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Émery, M. (2013). A Planar Borel Set Which Divides Every Non-negligible Borel Product. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_5
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DOI: https://doi.org/10.1007/978-3-319-00321-4_5
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