Topics in the Theory of Lifting
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 s...
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Format:  eBook 
Language:  English 
Published: 
Berlin, Heidelberg
Springer Berlin Heidelberg
1969, 1969

Edition:  1st ed. 1969 
Series:  Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics

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Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Table of Contents:
 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
 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 DunfordPettis 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
 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. Nonexistence 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?