On strong liftings for projective limits

Abstract

The main problem investigated in this paper is the following. Assume that we are given a convergent projective system of topological measure spaces ordered by ordinals. When does there exist a consistent system of liftings (densities, linear liftings) on the projective system converging to a lifting (density, linear lifting) on the limit space. We look mainly for strong or strong completion Baire liftings. We reduce the problem to the question about the existence of strong liftings being inverse images of other strong liftings under measure preserving mappings (Proposition 2.4) and then we adapt a condition applied earlier by A. and C. Ionescu Tulcea (14 )t o get as trong lifting for an arbitrary measure on a product space (Theorem 2.7). In this way we get some results (see Theorems 2.7, 5.3, 5.7, 6.4 and 6.5) extending the well known achievements of A. and C. Ionescu Tulcea (14 )a nd Fremlin (9). The application of projective limits allows us to carry over results obtained earlier only for product spaces (see e.g. (23), (18), (19), (20), (21)) to more general classes of topological probability spaces. In particular, we can extend the class of spaces for which there is a positive answer to a problem of J. Kupka (17 )c oncerning the permanence of the strong lifting property under the formation of products (see Theorem 6.5).