This article discusses some of the more recent results concerning strong liftings and Bore1 liftings, as well as the applications of such liftings to various topics in analysis, particularly derivation basis and the disintegration of measures. Section 2 introduces the basic notations and definitions in the theory of lifting and sketches its historical development.
Section 3 develops the basic results on strong liftings and Section 4, those on Bore1 liftings. Section 5 applies the preceding results to the study of derivation basis, which Section 6 involves the applications to the disintegration of measures. Section 7 summarizes the material presented while discussing some of the open problems in the theory of lifting.
SECTION 2
Let 2 be a locally compact space. We denote by d(Z) the vector space of all Radon measures on 2 and by d+(Z) the cone of all positive elements of d(Z). If p E M(Z) we denote by M* the set of all bounded p measurable fun&ions on Z to R. We write g for the family of p measurable subsets of Z and a0 for the subset of 9 consisting of measurable sets of jinite measure.
A mapping p: M*) +-M* is said to be a lifting of M* if (9 P(f 1 =f;
(ii) f = g implies p(f) = p(g);
(iii) p( 1) = 1; and p(0) = 0;
(iv) f > 0 implies p(f) > 0;
* Supported in part by the Loyola University Committee on Research.