A note on the Sierpiński partition (Q2785594)

For the Sierpiński partition \(A=\{(\xi,\zeta): \xi\leq\zeta<\omega_1\}\), \(B=\{(\xi,\zeta):\omega_1>\xi>\zeta\}\) of \(\omega_1\times\omega_1\) and the Dieudonné measure \(\lambda\) on \(\omega_1\) it is shown that \(A\), \(B\) are non-measurable with respect to \(\lambda\otimes\lambda\) in (ZF)\&(DC), whereas it is impossible to prove in (ZF)\&(DC) the existence of a non-measurable subset of \(\omega_1\) with respect to \(\lambda\). Furthermore, in (ZF)\&(DC) both sets \(A\) and \(B\) have the property of Baire in the product space \((\omega_1,\tau)\times (\omega_1,\tau)\) for a suitable topology \(\tau\) of \(\omega_1\), whereas the existence of a subset of \((\omega_1,\tau)\) without the Baire property cannot be established in (ZF)\&(DC). Finally, it is proved that in (ZF)\&(DC) one cannot conclude from the existence of some subset of a topological space without the property of Baire that there exists a subset of the corresponding topological product space without the property of Baire. However, one can show in (ZF)\&(DC) that the existence of a non-measurable subset with respect to some \(\sigma\)-finite measure \(\mu\) implies the existence of some non-measurable subset of the corresponding measure-theoretical product with respect to \(\mu\otimes\mu\).