Some theorems related to a problem of Ruziewicz

Jan Mycielski, Grzegorz Tomkowicz Bulletin Polish Acad. Sci. Math. MSC: Primary 03E05; Secondary 28C15, 46G12, 20E05, 51M05 DOI: 10.4064/ba231023-25-12 Published online: 17 January 2024


We apply deep results of Dougherty and Foreman and of Drinfeld and Margulis to give a very simple proof of the following theorem. Let $\mathbf B$ be the Boolean ring of Lebesgue measurable sets with the property of Baire in the sphere $\mathbf S^n$ or bounded Lebesgue measurable sets with the property of Baire in the Euclidean space $\mathbb R^{n+1}$ ($n \geq 2$). Then the Lebesgue measure in $\mathbf B$ is the unique finitely additive measure suitably normalized and invariant under isometries. Moreover, we prove that there exist everywhere dense $\mathbf G_{\delta}$ sets in the sphere $\mathbb S^n$ ($n \geq 2$) and in the cube $[0,1]^n$ ($n \geq 3$) that can be packed into arbitrarily small open sets using only subdivisions into finitely many Borel pieces.


  • Jan MycielskiDepartment of Mathematics
    University of Colorado
    Boulder, Colorado 80309-0395, USA
  • Grzegorz TomkowiczCentrum Edukacji $G^2$
    41-902 Bytom, Poland

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