Some theorems related to a problem of Ruziewicz
Tom 72 / 2024
Bulletin Polish Acad. Sci. Math. 72 (2024), 1-5
MSC: Primary 03E05; Secondary 28C15, 46G12, 20E05, 51M05
DOI: 10.4064/ba231023-25-12
Opublikowany online: 17 January 2024
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 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.
