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...

Full description

Main Authors: Ionescu Tulcea, Alexandra, Ionescu Tulcea, C. (Author)
Corporate Author: SpringerLink (Online service)
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
Subjects:
Online Access:
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 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
  • 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?