Title page
Preface
Chapter I. Measure and integration
1. The upper integral
2. The spaces L^p and L^p (l≤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
Chapter 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
Chapter 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
Chapter 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 10calizabiIity with the existence of a lifting of M^∞
4. Non-existence of a linear lifting for the L^p spaces (1≤p<∞)
5. The extension of a lifting to functions with values in a completely regular space
Chapter 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
Chapter VI. Integrability and measurability for abstract valued functIons
1. The spaces L^p_E and L^p_E (1≤p<∞)
2. Measurable functions
3. Further definitions and properties. The spaces L^∞_E and L^∞_E
4. The spaces M^∞_F[G] and L^∞_F[G]
5. The case of the spaces M^∞_F[F] and L^∞_F[F]
6. The spaces L^p_E[E] and L^p_E[E] (1≤p <∞)
7. A remark concerning the space M^∞_F[G]
Chapter VII. Various applications
1. An integral representation theorem
2. The existence of a linear lifting of M^∞_E is equivalent to the Dunford-Pettis theorem
3. Remarks concerning measurable functions and the spaces M^∞_E[E'] and L^∞_E[E']
4. The dual of L¹_E
5. The dual of L^p_E (1
6. A theorem of Strassen
7. An application to stochastic processes
Chapter 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
Chapter 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,M_+(S),µ) and F^∞(T,M_+(S),µ)
4. Integration of measures
5. Disintegration of measures. The general case
Chapter X. On certain endomorphisms of L^∞_R(Z,µ)
1. The spaces R(T₁,T₂))
2. The sets U(T₁,T₂) and the mappings β_u
3. The first main theorem
4. The spaces U*(T₁,T₂)
5. A condition equivalent with the strong lifting property
Appendix I. Some ergodic theorems
Appendix II. Notation and terminology
Open Problems
Bibliography
Subject Index
List of Symbols
