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On microscopic sets and Fubini Property in all directions

  • Adam Paszkiewicz EMAIL logo
Published/Copyright: October 20, 2018
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 68 Issue 5

Abstract

For the σ-ideal N of nullsets and σ-ideal M of microscopic sets, it was recently obtained that there exists a Borel set E⊂ℝ2 with the following property: Ex∈M for any x∈ℝ and {y;Ey∉N}∉M, for vertical sections Ex={y;(x,y)∈E} and horizontal sections Ey={x;(x,y)∈E} for E⊂ℝ2. Thus (N,M) does not satisfy Fubini Property. In this paper we obtain such Borel set E, that {y;Ey∉N}∉M and all non-horizontal (in a natural sense) sections of E are in M. Other Fubini type properties, with conditions written for all directions are also discussed.

MSC 2010: Primary 28A05; Secondary 03E15; 26A30
Keywords: nullsets; microscopic sets; Fubini property; marginal distributions

  1. Communicated by David Buhagiar

Acknowledgement

The author would like to thank the paper’s reviewer, whose remarks have been substantial in obtaining the final version of the article.

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Received: 2016-11-25
Accepted: 2017-07-07
Published Online: 2018-10-20
Published in Print: 2018-10-25

© 2018 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0165
Keywords for this article
nullsets; microscopic sets; Fubini property; marginal distributions

Articles in the same Issue

  1. Prof. RNDr. Beloslav Riečan, DrSc., Dr.h.c. *NOV. 10, 1936 – †AUG. 13, 2018
  2. On the n × n × n Rubik's Cube
  3. Congruences involving alternating harmonic sums modulo pαqβ
  4. On the proximity of large primes
  5. On the factorizations of cubic polynomials with the same discriminant modulo a prime
  6. Some results on abstract convexity of functions
  7. Remarks on b-Metric and metric-preserving functions
  8. Generalization of Ostrowski inequality for convex functions
  9. On microscopic sets and Fubini Property in all directions
  10. New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator
  11. Application of tan(φ(ξ)/2)-expansion method to burgers and foam drainage equations
  12. Approximate sentinels for diffusion phenomena with pollution
  13. Remarks on a semilinear system in ℝn motivated by difference equations
  14. Oscillation tests for difference equations with several non-monotone deviating arguments
  15. On the regularity of one-sided fractional maximal functions
  16. A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space
  17. A note about Volterra operator
  18. Some reverse and numerical radius inequalities
  19. A class of four-dimensional CR submanifolds in six dimensional nearly Kähler manifolds
  20. Sequential decreasing strong size properties
  21. Approximation of Information Divergences for Statistical Learning with Applications
  22. Wigner's theorem for an infinite set
  23. A note on automorphisms of lie ideals in prime rings
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