  Consider the extended hull of a weak model set together with its natural
shift action. Equip the extended hull with the Mirsky measure, which is a
certain natural pattern frequency measure. It is known that the extended hull
is a measure-theoretic factor of some group rotation, which is called the
underlying torus. Among other results, in the article "Periods and factors of
weak model sets" we showed that the extended hull is isomorphic to a factor
group of the torus, where certain periods of the window of the weak model set
have been factored out. This was proved for weak model sets having a compact
window. In this note, we argue that the same results hold for arbitrary
measurable and relatively compact windows. Our arguments crucially rely on
Moody's work on uniform distribution in model sets. We also discuss
implications for the diffraction of such weak model sets and discuss a new
class of examples which are generic for the Mirsky measure.
