Some theorems related to a problem of Ruziewicz
Streszczenie
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.
