Remarks on the strong lifting property for products

The class of compact measure spaces which possesses the attribute of having products with the strong lifting property is much larger than that of the metric spaces. This class includes every homogeneous space equipped with a quasi-invariant measure. This result, in conjunction with Losert’s example and Kupka’s arguments, yields invariant measures on transformation groups, which fail to have a lifting commuting with left translations. In addition, the previously mentioned class contains every product measure on an arbitrary product of metrizable spaces.