Productivity versus weak Fubini property (Q858080)

The Cantor space \(2^\omega\) is the set of all functions from \(\omega\) (\(=\) set of all nonnegative integers) into \(\{0,1\}\). Its subset INJ denotes the set of injective functions. For \(\varphi \in \text{INJ}\) a corresponding function \(\Phi:2^\omega \to 2^\omega\) is defined by \(\Phi(x)=x\circ \varphi\). If \(\mathfrak A\) is a family of subsets of \(2^\omega\) the authors define \(\omega(\mathfrak A):=\{A\subseteq 2^\omega \mid \exists \varphi\in \text{INJ}\;\text{with}\;\Phi[A]\in\mathfrak A\}\). Let \(\mathfrak J\) be a \(\sigma\)-ideal of subsets of \(2^\omega\). Then \(\mathfrak J\) is called productive if \(\omega(\mathfrak J)\subseteq \mathfrak J\). And \(\mathfrak J\) is said to have the weak Fubini property WFP, if for every \(A\subseteq 2^\omega\) and \(\varphi\in\text{INJ}\) if \(\Phi^{-1}[A]\) is in \(\mathfrak J\) then is \(A\). Then the authors prove: There exists a productive \(\sigma\)-ideal of subsets of the Cantor space \(2^\omega\) that does not have WFP.