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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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V: 10, 87-102, LNM 191 (1971)

**DELLACHERIE, Claude**

Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (Descriptive set theory)

Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections

Comment: See Dellacherie-Meyer,*Probabilités et Potentiel,* Chap. XI

Keywords: Analytic sets, Random sets, Section theorems

Nature: New exposition of known results

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V: 11, 103-126, LNM 191 (1971)

**DELLACHERIE, Claude**

Ensembles pavés et rabotages (Descriptive set theory)

A systematic study of the ``rabotages de Sierpinski'', used in Dellacherie 306 to solve several problems in probabilistic potential theory. The main paper on this subject

Comment: See Dellacherie,*Capacités et Processus Stochastiques,* 1970. Author should be consulted on recent developments (see 1526)

Keywords: Analytic sets, Capacities, Sierpinski's ``rabotages''

Nature: Original

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VII: 03, 33-35, LNM 321 (1973)

**DELLACHERIE, Claude**

Un crible généralisé (Descriptive set theory)

Given a Borel set $A$ in the product $E\times F$ of two compact metric sets, the set of all $x\in E$ such that the section $A(x)\subset Y$ is of second category is analytic

Comment: The authour discovered later that the main result is in fact due to Novikov: two references are given in 1252

Keywords: Analytic sets

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: 18, 406-407, LNM 465 (1975)

**DELLACHERIE, Claude**

Une remarque sur les espaces sousliniens de Bourbaki (Descriptive set theory)

The paper claims to prove that given any space with a Souslin topology i.e., which is Hausdorff and a continuous image of a Polish space, it is possible to strengthen the topology so that it becomes metrizable without losing the Souslin property. The author discovered later a serious mistake in his proof

Comment: The problem is still open, and interesting

Keywords: Analytic sets

Nature: False

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X: 28, 540-543, LNM 511 (1976)

**MOKOBODZKI, Gabriel**

Démonstration élémentaire d'un théorème de Novikov (Descriptive set theory)

Novikov's theorem asserts that any sequence of analytic subsets of a compact metric space with empty intersection can be enclosed in a sequence of Borel sets with empty intersection. This result has important consequences in descriptive set theory (see Dellacherie 915). A fairly simple proof of this theorem is given, which relates it to the first separation theorem (rather than the second separation theorem as it used to be)

Comment: Dellacherie in this volume (1032) further simplifies the proof. For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **9**

Keywords: Analytic sets

Nature: Original

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X: 29, 544-544, LNM 511 (1976)

**DELLACHERIE, Claude**

Correction à des exposés de 1973/74 (Descriptive set theory)

Corrections to 915 and 918

Keywords: Analytic sets, Semi-polar sets, Suslin spaces

Nature: Original

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X: 32, 579-593, LNM 511 (1976)

**DELLACHERIE, Claude**

Compléments aux exposés sur les ensembles analytiques (Descriptive set theory)

A new proof of Novikov's theorem (see 1028 and the corresponding comments) is given in the form of a Choquet theorem for multicapacities (with infinitely many arguments). Another (unrelated) result is a complement to 919 and 920, which study the space of stopping times. The language of stopping times is used to prove a deep section theorem due to Kondo

Keywords: Analytic sets, Section theorems, Capacities

Nature: Original

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XII: 46, 707-738, LNM 649 (1978)

**DELLACHERIE, Claude**

Théorie unifiée des capacités et des ensembles analytiques (Descriptive set theory)

A Choquet capacity takes one set as argument and produces a number. Along the years, one has considered multicapacities (which take as arguments finitely many sets) and capacitary operators (which produce sets instead of numbers). The essential result of this paper is that, if one allows functions of infinitely many arguments which produce sets, then the corresponding ``Choquet theorem'' gives all the classical results at a time, without need of an independent theory of analytic sets

Comment: For a more systematic exposition, see Chapter XI of Dellacherie-Meyer*Probabilités et Potentiel*

Keywords: Capacities, Analytic sets

Nature: Original

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XII: 52, 740-740, LNM 649 (1978)

**DELLACHERIE, Claude**

Correction à ``Un crible généralisé'' (Descriptive set theory)

Acknowledgement of priority and references concerning the result in 703

Keywords: Analytic sets

Nature: Correction

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XII: 55, 746-756, LNM 649 (1978)

**DELLACHERIE, Claude**

Quelques exemples familiers, en probabilités, d'ensembles analytiques non boréliens (Descriptive set theory, General theory of processes)

There is a tendency to consider that the naive, healthy probabilist should keep away from unnecessary abstraction, and in particular from analytic sets which are not Borel. This paper shows that such sets crop into probability theory in the most natural way. For instance, while the sample space of right-continuous paths with left limits is Borel, that of right-continuous paths without restriction on the left is coanalytic and non-Borel. Also, on the Borel sample space of right-continuous paths with left limits, the hitting time of a closed set is a function which is coanalytic and non-Borel

Keywords: Analytic sets

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

V: 10, 87-102, LNM 191 (1971)

Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (Descriptive set theory)

Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections

Comment: See Dellacherie-Meyer,

Keywords: Analytic sets, Random sets, Section theorems

Nature: New exposition of known results

Retrieve article from Numdam

V: 11, 103-126, LNM 191 (1971)

Ensembles pavés et rabotages (Descriptive set theory)

A systematic study of the ``rabotages de Sierpinski'', used in Dellacherie 306 to solve several problems in probabilistic potential theory. The main paper on this subject

Comment: See Dellacherie,

Keywords: Analytic sets, Capacities, Sierpinski's ``rabotages''

Nature: Original

Retrieve article from Numdam

VII: 03, 33-35, LNM 321 (1973)

Un crible généralisé (Descriptive set theory)

Given a Borel set $A$ in the product $E\times F$ of two compact metric sets, the set of all $x\in E$ such that the section $A(x)\subset Y$ is of second category is analytic

Comment: The authour discovered later that the main result is in fact due to Novikov: two references are given in 1252

Keywords: Analytic sets

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: 18, 406-407, LNM 465 (1975)

Une remarque sur les espaces sousliniens de Bourbaki (Descriptive set theory)

The paper claims to prove that given any space with a Souslin topology i.e., which is Hausdorff and a continuous image of a Polish space, it is possible to strengthen the topology so that it becomes metrizable without losing the Souslin property. The author discovered later a serious mistake in his proof

Comment: The problem is still open, and interesting

Keywords: Analytic sets

Nature: False

Retrieve article from Numdam

X: 28, 540-543, LNM 511 (1976)

Démonstration élémentaire d'un théorème de Novikov (Descriptive set theory)

Novikov's theorem asserts that any sequence of analytic subsets of a compact metric space with empty intersection can be enclosed in a sequence of Borel sets with empty intersection. This result has important consequences in descriptive set theory (see Dellacherie 915). A fairly simple proof of this theorem is given, which relates it to the first separation theorem (rather than the second separation theorem as it used to be)

Comment: Dellacherie in this volume (1032) further simplifies the proof. For a presentation in book form, see Dellacherie-Meyer,

Keywords: Analytic sets

Nature: Original

Retrieve article from Numdam

X: 29, 544-544, LNM 511 (1976)

Correction à des exposés de 1973/74 (Descriptive set theory)

Corrections to 915 and 918

Keywords: Analytic sets, Semi-polar sets, Suslin spaces

Nature: Original

Retrieve article from Numdam

X: 32, 579-593, LNM 511 (1976)

Compléments aux exposés sur les ensembles analytiques (Descriptive set theory)

A new proof of Novikov's theorem (see 1028 and the corresponding comments) is given in the form of a Choquet theorem for multicapacities (with infinitely many arguments). Another (unrelated) result is a complement to 919 and 920, which study the space of stopping times. The language of stopping times is used to prove a deep section theorem due to Kondo

Keywords: Analytic sets, Section theorems, Capacities

Nature: Original

Retrieve article from Numdam

XII: 46, 707-738, LNM 649 (1978)

Théorie unifiée des capacités et des ensembles analytiques (Descriptive set theory)

A Choquet capacity takes one set as argument and produces a number. Along the years, one has considered multicapacities (which take as arguments finitely many sets) and capacitary operators (which produce sets instead of numbers). The essential result of this paper is that, if one allows functions of infinitely many arguments which produce sets, then the corresponding ``Choquet theorem'' gives all the classical results at a time, without need of an independent theory of analytic sets

Comment: For a more systematic exposition, see Chapter XI of Dellacherie-Meyer

Keywords: Capacities, Analytic sets

Nature: Original

Retrieve article from Numdam

XII: 52, 740-740, LNM 649 (1978)

Correction à ``Un crible généralisé'' (Descriptive set theory)

Acknowledgement of priority and references concerning the result in 703

Keywords: Analytic sets

Nature: Correction

Retrieve article from Numdam

XII: 55, 746-756, LNM 649 (1978)

Quelques exemples familiers, en probabilités, d'ensembles analytiques non boréliens (Descriptive set theory, General theory of processes)

There is a tendency to consider that the naive, healthy probabilist should keep away from unnecessary abstraction, and in particular from analytic sets which are not Borel. This paper shows that such sets crop into probability theory in the most natural way. For instance, while the sample space of right-continuous paths with left limits is Borel, that of right-continuous paths without restriction on the left is coanalytic and non-Borel. Also, on the Borel sample space of right-continuous paths with left limits, the hitting time of a closed set is a function which is coanalytic and non-Borel

Keywords: Analytic sets

Nature: Original

Retrieve article from Numdam