On the Lifting Property and Disintegration of Measures

In what follows we shall present certain results concerning the lifting property of a space L°°(Z, ju), where Z is locally compact and 11 a positive measure on Z. We shall discuss several applications and state several problems. Most of the results which will be presented here were jointly obtained by A. lonescu Tulcea and the author, during the last few years. Some of these results have already appeared in print while others have not yet been published. The presentation will be divided as follows: 1. The existence of a lifting. 2. The strong lifting. 3. Liftings commuting with certain groups of mappings. 4. Lower densities and liftings. 5. Integral representations of operators. 6. Disintegration of measures. 7. Point realizations of endomorphisms of L*>(Z, n) spaces. 8. Final remarks. Notations. Here are some of the notations which will be used below. For a locally compact space X denote by M(X) the vector space of all (Radon) measures on X and by M+(X) the cone of all positive measures on X. For every (3ÇzM+(X) denote by MTM(X, /3) the algebra of all ƒ : X—*R which are bounded and /3-measurable and by N«>(X, 0) the ideal of all fEM(Xt /3) which are locally /3-negligible. By L°°(X, j8) we shall denote the quotient algebra M°°(X, P)/N"(X, j8) and by ir:f—*f the canonical mapping of Af°°(X, j3) onto L°°(X, j3). For fÇzM^^X, j8) and gÇzM°°(X, j3) we write / = g whenever / = | (therefore ƒ s g means that ƒ and g coincide locally almost everywhere, while ƒ = g means that ƒ and g coincide everywhere). We denote by C^iX) the subalgebra of ilf°°(X, |8) consisting of all continuous functions ƒ G M°°(X, j3) and by 3Z(X) the subalgebra of C^iX) consisting of all continuous functions with compact support. For g£3C(X) and /3GM(Z) write