Author	Tulcea, A. Ionescu. author
Title	Topics in the Theory of Lifting [electronic resource] / by A. Ionescu Tulcea, C. Ionescu Tulcea
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 1969
Connect to	http://dx.doi.org/10.1007/978-3-642-88507-5
Descript	X, 192 p. 1 illus. online resource

The problem as to whether or not there exists a lifting of the M't/. 1 space ) corresponding to the real line and Lebesgue measure on it was first raised by A. Haar. It was solved in a paper published in 1931 [102] by 1. von Neumann, who established the existence of a lifting in this case. In subsequent papers J. von Neumann and M. H. Stone [105], and later on 1. Dieudonne [22], discussed various algebraic aspects and generalizations of the problem. Attemps to solve the problem as to whether or not there exists a lifting for an arbitrary M't/. space were unsuccessful for a long time, although the problem had significant connections with other branches of mathematics. Finally, in a paper published in 1958 [88], D. Maharam established, by a delicate argument, that a lifting of M't/. always exists (for an arbiยญ trary space of a-finite mass). D. Maharam proved first the existence of a lifting of the M't/. space corresponding to a product X = TI {ai,b,} ieI and a product measure J.1= Q9 J.1i' with J.1i{a;}=J.1i{b,}=! for all iE/. ,eI Then, she reduced the general case to this one, via an isomorphism theorem concerning homogeneous measure algebras [87], [88]. A different and more direct proof of the existence of a lifting was subsequently given by the authors in [65]' A variant of this proof is presented in chapter 4

I. Measure and integration -- 1. The upper integral -- 2. The spaces ?p and Lp (1 ? p < + ?) -- 3. The integral -- 4. Measurable functions -- 5. Further definitions and properties of measurable functions and sets -- 6. Carathรฉodory measure -- 7. The essential upper integral. The spaces M? and L? -- 8. Localizable and strictly localizable spaces -- 9. The case of abstract measures and of Radon measures -- II. Admissible subalgebras and projections onto them -- 1. Admissible subalgebras -- 2. Multiplicative linear mappings -- 3. Extensions of linear mappings -- 4. Projections onto admissible subalgebras -- 5. Increasing sequences of projections corresponding to admissible subalgebras -- III. Basic definitions and remarks concerning the notion of lifting -- 1. Linear liftings and liftings of an admissible subalgebra. Lower densities -- 2. Linear liftings, liftings and extremal points -- 3. On the measurability of the upper envelope. A limit theorem -- IV. The existence of a lifting -- 1. Several results concerning the extension of a lifting -- 2. The existence of a lifting of M? -- 3. Equivalence of strict localizability with the existence of a lifting of M? -- 4. Non-existence of a linear lifting for the ?p spaces (1 ? p < ?) -- 5. The extension of a lifting to functions with values in a completely regular space -- V. Topologies associated with lower densities and liftings -- 1. The topology associated with a lower density -- 2. Construction of a lifting from a lower density using the density topology -- 3. The topologies associated with a lifting -- 4. An example -- 5. Liftings compatible with topologies -- 6. A remark concerning liftings for functions with values in a completely regular space -- VI. Integrability and measurability for abstract valued functions -- 1. The spaces ?EP and LEP (1 ? p < + ?) -- 2. Measurable functions -- 3. Further definitions and properties. The spaces ?E? and LE? -- 4. The spaces MF? [G] and LF? [G] -- 5. The case of the spaces ME? [E] and LE? [E] -- 6. The spaces ?EP [E] and LEP [E] (1 ? p < + ?) -- 7. A remark concerning the space MF? [G] -- VII. Various applications -- 1. An integral representation theorem -- 2. The existence of a linear lifting of MR? is equivalent to the Dunford-Pettis theorem -- 3. Remarks concerning measurable functions and the spaces ME? [E?] and LE? [E?] -- 4. The dual of LE1 -- 5. The dual of LEP (1 < p < + ?) -- 6. A theorem of Strassen -- 7. An application to stochastic processes -- VIII. Strong liftings -- 1. The notion of strong lifting -- 2. Further results concerning strong liftings. Examples -- 3. An example and several related results -- 4. The notion of almost strong lifting -- 5. The notions of almost strong and strong lifting for topological spaces -- Appendix. Borel liftings -- IX. Domination of measures and disintegration of measures -- 1. Convex cones of continuous functions and the domination of measures -- 2. Disintegration of measures. The case of a compact space and a continuous mapping -- 3. The cones F (T, ?+(S), ยต) and F? (T, ?+(S), ยต) -- 4. Integration of measures -- 5. Disintegration of measures. The general case -- X. On certain endomorphisms of LR?(Z, ยต) -- 1. The spaces R(I1, I2) -- 2. The sets U(I1, I2) and the mappings ?u -- 3. The first main theorem -- 4. The spaces U*(I1, I2) -- 5. A condition equivalent with the strong lifting property -- Appendix I. Some ergodic theorems -- Appendix II. Notation and terminology -- Open Problems -- List of Symbols