III: 05, 97-114, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires I (
Descriptive set theory,
Markov processes,
General theory of processes)
A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972
Keywords: Sets with countable sectionsNature: Original
Retrieve article from Numdam
III: 06, 115-136, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires II (
Descriptive set theory,
Markov processes)
Among the many proofs that an uncountable Borel set of the line contains a perfect set, a proof of Sierpinski (
Fund. Math.,
5, 1924) can be extended to an abstract set-up to show that a non-semi-polar Borel set contains a non-semi-polar compact set
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972. More recent proofs no longer depend on ``rabotages'': Dellacherie-Meyer,
Probabilités et potentiel, Appendix to Chapter IV
Keywords: Sierpinski's ``rabotages'',
Semi-polar setsNature: Original Retrieve article from Numdam
V: 07, 77-81, LNM 191 (1971)
DELLACHERIE, Claude
Quelques commentaires sur les prolongements de capacités (
Descriptive set theory)
Remarks on the extension of capacities from sets to functions. Probably superseded by the work of Mokobodzki on functional capacities
Comment: See Dellacherie-Meyer,
Probabilités et Potentiel, Chap. XI: capacités fonctionnelles
Keywords: CapacitiesNature: Original Retrieve article from Numdam
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 theoremNature: Original Retrieve article from Numdam
V: 09, 86-86, LNM 191 (1971)
DELLACHERIE, Claude
Correction à ``Ensembles Aléatoires II'' (
Descriptive set theory)
Correction to Dellacherie
306Comment: See Dellacherie
511Keywords: Sierpinski's ``rabotages''Nature: Original Retrieve article from Numdam
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 theoremsNature: New exposition of known results Retrieve article from Numdam
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 Retrieve article from Numdam
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
1252Keywords: Analytic setsNature: 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 theoremNature: 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 theoremNature: 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 timesNature: Original Retrieve article from Numdam
IX: 17, 390-405, LNM 465 (1975)
DELLACHERIE, Claude
Jeux infinis avec information complète et temps d'arrêt (
Descriptive set theory)
This is a sequel to the preceding paper
916. It shows how well the language of stopping times applies, not only to the second separation theorem, but to the Gale-Stewart theorem on the determinacy of open games
Comment: The original remark on the relation between game determinacy and separation theorems, due to Blackwell (1967), led to a huge literature. More details can be found in chapter XXIV of Dellacherie-Meyer,
Probabilités et potentiel Keywords: Determinacy of games,
Gale and Stewart theoremNature: Original Retrieve article from Numdam
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 setsNature: False Retrieve article from Numdam
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
9Keywords: Analytic setsNature: Original Retrieve article from Numdam
X: 29, 544-544, LNM 511 (1976)
DELLACHERIE, Claude
Correction à des exposés de 1973/74 (
Descriptive set theory)
Corrections to
915 and
918Keywords: Analytic sets,
Semi-polar sets,
Suslin spacesNature: Original Retrieve article from Numdam
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,
CapacitiesNature: Original Retrieve article from Numdam
XI: 04, 34-46, LNM 581 (1977)
DELLACHERIE, Claude
Les dérivations en théorie descriptive des ensembles et le théorème de la borne (
Descriptive set theory)
At the root of set theory lies Cantor's definition of the ``derived set'' $\delta A$ of a closed set $A$, i.e., the set of its non-isolated points, with the help of which Cantor proved that a closed set can be decomposed into a perfect set and a countable set. One may define the index $j(A)$ to be the smallest ordinal $\alpha$ such that $\delta^\alpha A=\emptyset$, or $\omega_1$ if there is no such ordinal. Considering the set $F$ of all closed sets as a (Polish) topological space, ordered by inclusion, $\delta$ as an increasing mapping from $F$ such that $\delta A\subset A$, let $D$ be the set of all $A$ such that $j(A)<\omega_1$ (thus, the set of all countable closed sets). Then $D$ is coanalytic and non-Borel, while the index is bounded by a countable ordinal on every analytic subset of $D$. These powerful results are stated abstractly and proved under very general conditions. Several examples are given
Comment: See a correction in
1241, and several examples in Hillard
1242. The whole subject has been exposed anew in Chapter~XXIV of Dellacherie-Meyer,
Probabilités et PotentielKeywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Exposition,
Original additions Retrieve article from Numdam
XII: 41, 523-523, LNM 649 (1978)
DELLACHERIE, Claude
Erratum et addendum à ``les dérivations en théorie descriptive des ensembles et le théorème de la borne'' (
Descriptive set theory)
A few corrections to
1104Keywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Correction Retrieve article from Numdam
XII: 42, 524-563, LNM 649 (1978)
HILLARD, Gérard
Exemples de normes en théorie descriptive des ensembles (
Descriptive set theory)
The situations described in this paper are special cases of
1104, where a coanalytic set $A$ was represented as the union of an increasing family $A_{\alpha}$ of analytic sets indexed by the countable ordinals, such that every analytic subset of $A$ is contained in some $A_{\alpha}$. The hypotheses of
1104 are not easy to check: they are shown here to include the classical Cantor derivation on the coanalytic space of countable compact sets, and a new example on the coanalytic space of all right continuous functions
Comment: The whole subject has been exposed anew in Chapter XXIV of Dellacherie-Meyer,
Probabilités et PotentielKeywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Original Retrieve article from Numdam
XII: 43, 564-566, LNM 649 (1978)
DELLACHERIE, Claude;
MOKOBODZKI, Gabriel
Deux propriétés des ensembles minces (abstraits) (
Descriptive set theory)
Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets
Keywords: Thin sets,
Semi-polar sets,
Essential supremaNature: Original Retrieve article from Numdam
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 PotentielKeywords: Capacities,
Analytic setsNature: Original Retrieve article from Numdam
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
703Keywords: Analytic setsNature: Correction Retrieve article from Numdam
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 setsNature: Original Retrieve article from Numdam
