Completion regular measures on product spaces with application to the existence of Baire strong liftings
This paper deals with the problem of the existence of Baire strong liftings for measures supported by products of compact metrizable spaces. Only a few results are known in this field as yet. Since such measures must necessarily be completion regular, the permanence of the completion regularity under products is also discussed. The results given in this paper are the following ones: 1. Assuming the continuum hypothesis any completion regular measure of full support on a product of less or equal than c compact metric spaces admits a Baire strong lifting. This result has been generalized by the authors themselves (see `Measures on product spaces and the existence of strong Baire lifting', preprint). 2. Subject to Martin's Axiom and the negation of the continuum hypothesis, the Radon product measure of two completion regular measures, one on a compact space and the other one on an uncountable product of compact metric spaces is completion regular itself. This extends a result of \textit{J. R. Choksi} and \textit{D. H. Fremlin} on completion regularity in products [see Math. Ann. 241, 113-128 (1979; Zbl 0387.60006)] 3. Assuming the continuum hypothesis the Radon product measure of two completion regular measures each of which is supported on a product of less or equal than c compact metric spaces is completion regular and admits a Baire strong lifting.
scientific article; zbMATH DE number 4175308