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140122  eng 
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a 9783642885075

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1 

a Ionescu Tulcea, Alexandra

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a Topics in the Theory of Lifting
h Elektronische Ressource
c by Alexandra Ionescu Tulcea, C. Ionescu Tulcea

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a 1st ed. 1969

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a Berlin, Heidelberg
b Springer Berlin Heidelberg
c 1969, 1969

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a X, 192 p. 1 illus
b online resource

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a 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

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a 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 

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a 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 

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a 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? 

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a Probability Theory and Stochastic Processes

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a Probabilities

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a Ionescu Tulcea, C.
e [author]

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a SpringerLink (Online service)

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0 
7 
a eng
2 ISO 6392

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b SBA
a Springer Book Archives 2004

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a Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics

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u https://doi.org/10.1007/9783642885075?nosfx=y
x Verlag
3 Volltext

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a 519.2

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a 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 subsequent papers J. von Neumann and M. H. Stone [105], and later on 1. Dieudonne [22], discussed various algebraic aspects and generalizations of the problem. Attemps to solve the problem as to whether or not there exists a lifting for an arbitrary M't/. space were unsuccessful for a long time, although the problem had significant connections with other branches of mathematics. Finally, in a paper published in 1958 [88], D. Maharam established, by a delicate argument, that a lifting of M't/. always exists (for an arbi trary space of afinite mass). D. Maharam proved first the existence of a lifting of the M't/. space corresponding to a product X = TI {ai,b,} ieI and a product measure J.1= Q9 J.1i' with J.1i{a;}=J.1i{b,}=! for all iE/. ,eI Then, she reduced the general case to this one, via an isomorphism theorem concerning homogeneous measure algebras [87], [88]. A different and more direct proof of the existence of a lifting was subsequently given by the authors in [65]' A variant of this proof is presented in chapter 4
