Abstract
Our main inspiration is the work in paper (Gitik and Shelah in Isr J Math 124(1):221–242, 2001). We will prove that for a partition \({\mathcal{A}}\) of the real line into meager sets and for any sequence \({\mathcal{A}_n}\) of subsets of \({\mathcal{A}}\) one can find a sequence \({\mathcal{B}_n}\) such that \({\mathcal{B}_{n}}\)’s are pairwise disjoint and have the same “outer measure with respect to the ideal of meager sets”. We get also generalization of this result to a class of σ-ideals posessing Suslin property. However, in that case we use additional set-theoretical assumption about non-existing of quasi-measurable cardinal below continuum.
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References
Brzuchowski J., Cichoń J., Grzegorek E., Ryll-Nardzewski C.: On the existence of nonmeasurable unions. Bull. Polish Acad. Sci. Math. 27, 447–448 (1979)
Cichoń J., Morayne M., Rałowski R., Ryll-Nardzewski C., Żeberski S.: On nonmeasurable unions. Topol. Appl. 154, 884–893 (2007)
Erdös P.: Some remarks on set theory. Proc. Amer. Math. Soc. 1, 127–141 (1950)
Gitik M., Shelah S.: More on real-valued measurable cardinals and forcings with ideals. Isr. J. Math. 124(1), 221–242 (2001)
Gitik M., Shelah S.: Forcing with ideals and simple forcing notions. Isr. J. Math. 62, 129–160 (1989)
Jech T.: Set theory. Springer, Berlin (2000)
Rałowski R., Żeberski S.: Completely nonmeasurable union. Cent. Eur. J. Math. 8(4), 683–687 (2010)
Taylor A.: On saturated sets of ideals and Ulam’s problem. Fund. Math. 109, 37–53 (1980)
Zeberski S.: On completely nonmeasurable unions. Math. Log. Q. 53(1), 38–42 (2007)
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Żeberski, S. Inscribing nonmeasurable sets. Arch. Math. Logic 50, 423–430 (2011). https://doi.org/10.1007/s00153-010-0223-6
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DOI: https://doi.org/10.1007/s00153-010-0223-6