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4 matches found
V: 08, 82-85, LNM 191 (1971)
DELLACHERIE, Claude
Une démonstration du théorème de séparation des ensembles analytiques (Descriptive set theory)
The first separation theorem can be deduced from Choquet's capacity theorem
Comment: Starting point in Sion, Ann. Inst. Fourier, 13, 1963. This proof has become standard, see Dellacherie-Meyer, Probabilités et Potentiel, Chap. III
Keywords: Analytic sets, Capacities, Separation theorem
Nature: Original
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VII: 06, 48-50, LNM 321 (1973)
DELLACHERIE, Claude
Une démonstration du théorème de Souslin-Lusin (Descriptive set theory)
The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets
Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See Un cours sur les ensembles analytiques, in Analytic Sets, C.A. Rogers ed., Academic Press 1980
Keywords: Borel sets, Analytic sets, Separation theorem
Nature: New exposition of known results
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IX: 15, 336-372, LNM 465 (1975)
DELLACHERIE, Claude
Ensembles analytiques, théorèmes de séparation et applications (Descriptive set theory)
According to the standard (``first'') separation theorem, in a compact metric space or any space which is Borel isomorphic to it, two disjoint analytic sets can be separated by Borel sets, and in particular any bianalytic set (analytic and coanalytic i.e., complement of analytic) is Borel. Not so in general metric spaces. That the same statement holds in full generality with ``bianalytic'' instead of ``Borel'' is the second separation theorem, which according to the general opinion was considered much more difficult than the first. This result and many more (on projections of Borel sets with compact sections or countable sections, for instance) are fully proved in this exposition
Comment: See also the next paper 916, the set of lectures by Dellacherie in C.A. Rogers, Analytic Sets, Academic Press 1981, and chapter XXIV of Dellacherie-Meyer, Probabilités et potentiel
Keywords: Second separation theorem
Nature: Exposition
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IX: 16, 373-389, LNM 465 (1975)
DELLACHERIE, Claude; MEYER, Paul-André
Ensembles analytiques et temps d'arrêt (Descriptive set theory)
This is a sequel to the preceding paper 915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on \$N^N\$ is given a compact metric topology and the space of non-finite stopping times appears as the universal analytic, non-Borel set, from which all analytic sets can be constructed. Many proofs become very natural in this language
Comment: See also the next paper 917, the set of lectures by Dellacherie in C.A. Rogers, Analytic Sets, Academic Press 1981, and chapter XXIV of Dellacherie-Meyer, Probabilités et potentiel
Keywords: Second separation theorem, Stopping times
Nature: Original
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