Measures on product spaces and the existence of strong Baire liftings (Q1207667)

Some notions are introduced for studying measures on product spaces, the main concept being that of property \((*)\) ``For every uncountable family \(\{V_ \alpha\}\) of open sets in \(X=\prod_{i\in I}X_ i\), where the \(V_ \alpha\) depend on pairwise disjoint sets of coordinates, we have: \(\mu(\cup_ \alpha V_ \alpha)=1\)''. In case when the topological factors are separable metric spaces, this property is equivalent to the completion regularity. We prove that \((*)\) is preserved under arbitrary products of measure spaces. As a consequence, we deduce a series of related results in measure theory (some of which are known). In particular, the following extension of a result by Losert is obtained: Subject to CH, every product of \(\leq\aleph_ 2\) many completion regular measures, each supported on any product of \(\leq\aleph_ 1\) many compact metric spaces admits a strong Baire lifting.