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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 arbi 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.
Contents
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. Carathéodory 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 ?EPand 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), µ) and F? (T, ?+(S), µ).- 4. Integration of measures.- 5. Disintegration of measures. The general case.- X. On certain endomorphisms of LR?(Z, µ).- 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.
