Quick search | Browse volumes | |

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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,

Keywords: Analytic sets, Capacities, Separation theorem

Nature: Original

Retrieve article from Numdam

VII: 06, 48-50, LNM 321 (1973)

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

Keywords: Borel sets, Analytic sets, Separation theorem

Nature: New exposition of known results

Retrieve article from Numdam

IX: 15, 336-372, LNM 465 (1975)

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,

Keywords: Second separation theorem

Nature: Exposition

Retrieve article from Numdam

IX: 16, 373-389, LNM 465 (1975)

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 $

Comment: See also the next paper 917, the set of lectures by Dellacherie in C.A. Rogers,

Keywords: Second separation theorem, Stopping times

Nature: Original

Retrieve article from Numdam