Table of 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—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 shastic 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.