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 |
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