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I: 04, 52-53, LNM 39 (1967)

**DELLACHERIE, Claude**

Un complément au théorème de Weierstrass-Stone (Functional analysis)

An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore

Keywords: Stone-Weierstrass theorem

Nature: Original

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III: 04, 93-96, LNM 88 (1969)

**DELLACHERIE, Claude**

Une application aux fonctionnelles additives d'un théorème de Mokobodzki (Markov processes)

Mokobodzki showed the existence of ``rapid ultrafilters'' on the integers, with the property that applied to a sequence that converges in probability they converge a.s. (see for instance Dellacherie-Meyer,*Probabilité et potentiels,* Chap. II, **27**). They are used here to prove that every continuous additive functional of a Markov process has a ``perfect'' version

Comment: See also 203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

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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 sections

Nature: Original

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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 sets

Nature: Original

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IV: 05, 60-70, LNM 124 (1970)

**DELLACHERIE, Claude**

Un exemple de la théorie générale des processus (General theory of processes)

In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly

Comment: This example has become classical. See for example Dellacherie-Meyer,*Probabilités et Potentiel,* Chap IV. On the other hand, it can be extended to deal with (unmarked) point processes: see Chou-Meyer 906

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

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IV: 06, 71-72, LNM 124 (1970)

**DELLACHERIE, Claude**

Au sujet des sauts d'un processus de Hunt (Markov processes)

Two a.s. results on jumps: the process cannot jump*from * a semi-polar set; at the first hitting time of any finely closed set $F$, either the process does not jump, or it jumps from outside $F$

Comment: Both results are improvements of previous results of Meyer and Weil

Keywords: Hunt processes, Semi-polar sets

Nature: Original

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IV: 07, 73-75, LNM 124 (1970)

**DELLACHERIE, Claude**

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

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IV: 08, 76-76, LNM 124 (1970)

**DELLACHERIE, Claude**

Un lemme de théorie de la mesure (Measure theory)

A lemma used by Erdös, Kesterman and Rogers (*Coll. Math.,* **XI**, 1963) is reduced to the fact that a sequence of bounded r.v.'s contains a weakly convergent subsequence

Keywords: Convergence in norm, Subsequences

Nature: Original proofs

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IV: 19, 240-282, LNM 124 (1970)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**; **LETTA, Giorgio**; **MEYER, Paul-André**

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (*Comm. Pure Appl. Math.*, **22**, 1969), which constructs by a probability method a unique semigroup whose generator is an elliptic second order operator with continuous coefficients (the analytic approach either deals with operators in divergence form, or requires some Hölder condition). The contribution of G.~Letta nicely simplified the proof

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,*Multidimensional Diffusion Processes,* Springer 1979

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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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: Capacities

Nature: Original

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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: 09, 86-86, LNM 191 (1971)

**DELLACHERIE, Claude**

Correction à ``Ensembles Aléatoires II'' (Descriptive set theory)

Correction to Dellacherie 306

Comment: See Dellacherie 511

Keywords: Sierpinski's ``rabotages''

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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V: 12, 127-137, LNM 191 (1971)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**

Un contre-exemple au problème des laplaciens approchés (Martingale theory)

The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years

Comment: Problem originated in Meyer,*Ill. J. Math.*, **7**, 1963

Keywords: Submartingales, Supermartingales

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: 04, 36-37, LNM 321 (1973)

**DELLACHERIE, Claude**

Temps d'arrêt totalement inaccessibles (General theory of processes)

Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left

Keywords: Stopping times, Accessible sets, Totally inaccessible stopping times

Nature: Original

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VII: 05, 38-47, LNM 321 (1973)

**DELLACHERIE, Claude**

Sur les théorèmes fondamentaux de la théorie générale des processus (General theory of processes)

This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved

Comment: The idea of this paper has proved fruitful. See for instance Lenglart, 1449; Le Jan,*Z. für W-Theorie * **44**, 1978

Keywords: Stopping times, Section theorems

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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VII: 07, 51-57, LNM 321 (1973)

**DELLACHERIE, Claude**

Une conjecture sur les ensembles semi-polaires (Markov processes)

For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets

Comment: The conjecture can be proved, using a general result of Mokobodzki, see 1238

Keywords: Polar sets, Semi-polar sets

Nature: Original

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VII: 08, 58-60, LNM 321 (1973)

**DELLACHERIE, Claude**

Potentiels de fonctionnelles additives. Un contre-exemple de Knight (Markov processes)

An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point

Keywords: Additive functionals

Nature: Exposition, Original additions

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VIII: 04, 22-24, LNM 381 (1974)

**DELLACHERIE, Claude**

Un ensemble progressivement mesurable... (General theory of processes)

The set of starting times of Brownian excursions from $0$ is a well-known example of a progressive set which does not contain any graph of stopping time. Here it is shown that considering the same set for the excursions from any $a$ and taking the union of all $a$, the corresponding set has the same property and has uncountable sections

Comment: Other such examples are known, such as the set of times at which the law of the iterated logarithm fails

Keywords: Progressive sets, Section theorems

Nature: Original

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VIII: 05, 25-26, LNM 381 (1974)

**DELLACHERIE, Claude**

Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)

This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems

Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,*Z. für W-theorie,* **38**, 1977 and Yor 1221. For another approach to the restricted case considered here, see Ruiz de Chavez 1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property

Keywords: Brownian motion, Poisson processes, Previsible representation

Nature: Original

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IX: 08, 239-245, LNM 465 (1975)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

Un nouveau théorème de projection et de section (General theory of processes)

Optional section and projection theorems are proved without assuming the ``usual conditions'' on the filtration

Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie 705. The result takes its definitive form in Dellacherie-Meyer,*Probabilités et Potentiel,* theorems IV.84 of vol. A and App.1, \no~6

Keywords: Section theorems, Optional processes, Projection theorems

Nature: Original

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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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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 theorem

Nature: Original

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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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IX: 28, 494-494, LNM 465 (1975)

**DELLACHERIE, Claude**

Correction à ``Intégrales stochastiques par rapport...'' (General theory of processes)

This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie 805

Comment: Another way of filling this gap is given by Ruiz de Chavez 1821. The same gap for the Poisson process is corrected in 2002

Keywords: Previsible representation

Nature: Original

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IX: 29, 495-495, LNM 465 (1975)

**DELLACHERIE, Claude**

Une propriété des ensembles semi-polaires (Markov processes)

It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)

Keywords: Semi-polar 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: 30, 545-577, LNM 511 (1976)

**DELLACHERIE, Claude**

Sur la construction de noyaux boréliens (Measure theory)

This answers questions of Getoor 923 and Meyer 924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest

Comment: For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **41**. The hypothesis that the space is compact is sometimes troublesome for the applications

Keywords: Pseudo-kernels, Regularization

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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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 Potentiel*

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Exposition, Original additions

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XI: 05, 47-50, LNM 581 (1977)

**DELLACHERIE, Claude**

Deux remarques sur la séparabilité optionnelle (General theory of processes)

Optional separability was defined by Doob,*Ann. Inst. Fourier,* **25**, 1975. See also Benveniste, 1025. The main remark in this paper is the following: given any optional set $H$ with countable dense sections, there exists a continuous change of time $(T_t)$ indexed by $[0,1[$ such that $H$ is the union of all graphs $T_t$ for $t$ dyadic. Thus Doob's theorem amounts to the fact that every optional process becomes separable in the ordinary sense once a suitable continuous change of time has been performed

Keywords: Optional processes, Separability, Changes of time

Nature: Original

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XI: 22, 362-364, LNM 581 (1977)

**DELLACHERIE, Claude**

Sur la régularisation des surmartingales (Martingale theory)

It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)

Comment: See 1524

Keywords: General filtrations, Strong supermartingales

Nature: Original

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XI: 23, 365-375, LNM 581 (1977)

**DELLACHERIE, Claude**; **STRICKER, Christophe**

Changements de temps et intégrales stochastiques (Martingale theory)

A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135

Comment: The last section states an interesting open problem

Keywords: Changes of time, Spectral representation

Nature: Original

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XII: 10, 70-77, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

A propos du travail de Yor sur le grossissement des tribus (General theory of processes)

This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 12, 98-113, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**; **YOR, Marc**

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

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XII: 29, 424-424, LNM 649 (1978)

**DELLACHERIE, Claude**

Convergence en probabilité et topologie de Baxter-Chacón (General theory of processes)

It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability

Keywords: Stopping times

Nature: Original

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XII: 30, 425-427, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)

Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$

Keywords: Stopping times

Nature: Original

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XII: 38, 509-511, LNM 649 (1978)

**DELLACHERIE, Claude**

Appendice à l'exposé de Mokobodzki (Measure theory, General theory of processes)

Some comments on 1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from 707

Keywords: Sets with countable sections, Semi-polar sets

Nature: Original

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XII: 39, 512-514, LNM 649 (1978)

**DELLACHERIE, Claude**

Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (General theory of processes)

The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes

Keywords: Essential suprema, Evanescent sets

Nature: Original

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XII: 40, 515-522, LNM 649 (1978)

**DELLACHERIE, Claude**

Supports optionnels et prévisibles d'une P-mesure et applications (General theory of processes)

A $P$-measure is a measure on $\Omega\times**R**_+$ which does not charge $P$-evanescent sets. A $P$-measure has optional and previsible projections which are themselves $P$-measures. As usual, supports are minimal sets carrying a measure, possessing different properties like being optional/previsible, being right/left closed. The purpose of the paper is to find out which kind of supports do exist. Applications are given to honest times

Comment: See 1339 for a complement concerning honest times

Keywords: Projection theorems, Support, Honest times

Nature: Original

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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 1104

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Correction

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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 suprema

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: 54, 742-745, LNM 649 (1978)

**DELLACHERIE, Claude**

Quelques applications du lemme de Borel-Cantelli à la théorie des semimartingales (Martingale theory, Stochastic calculus)

The general idea is the following: many constructions relative to one single semimartingale---like finding a sequence of stopping times increasing to infinity which reduce a local martingale, finding a change of law which sends a given semimartingale into $H^1$ or $H^2$ (locally)---can be strengthened to handle at the same time countably many given semimartingales

Nature: Original

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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

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XIII: 31, 371-377, LNM 721 (1979)

**DELLACHERIE, Claude**

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,*Probabilités et Potentiels B,* Chapter VI

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

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XV: 24, 320-346, LNM 850 (1981)

**DELLACHERIE, Claude**; **LENGLART, Érik**

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (General theory of processes)

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

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XV: 26, 351-370, LNM 850 (1981)

**DELLACHERIE, Claude**

Mesurabilité des débuts et théorème de section~: le lot à la portée de toutes les bourses (General theory of processes)

One of the main topics in these seminars has been the application to stochastic processes of results from descriptive set theory and capacity theory, at different levels. Since these results are considered difficult, many attempts have been made to shorten and simplify the exposition. A noteworthy one was 511, in which Dellacherie introduced ``rabotages'' (306) to develop the theory without analytic sets; see also 1246, 1255. The main feature of this paper is a new interpretation of rabotages as a two-persons game, ascribed to Telgarsky though no reference is given, leading to a pleasant exposition of the whole theory and its main applications

Keywords: Section theorems, Capacities, Sierpinski's ``rabotages''

Nature: Original

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XV: 27, 371-387, LNM 850 (1981)

**DELLACHERIE, Claude**

Sur les noyaux $\sigma$-finis (Measure theory)

This paper is an improvement of 1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called*measurability * of $m$) is the following: for every auxiliary space $(Z, {\cal Z})$, the mapping $(x,z)\mapsto m_x\otimes \epsilon_z$ is again a kernel (in fact, the auxiliary space $**R**$ is all one needs). The case of ``basic'' kernels, considered in 1235, is thoroughly discussed

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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XVI: 02, 8-28, LNM 920 (1982)

**DELLACHERIE, Claude**; **FEYEL, Denis**; **MOKOBODZKI, Gabriel**

Intégrales de capacités fortement sous-additives

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XVI: 03, 29-40, LNM 920 (1982)

**DELLACHERIE, Claude**

Appendice à l'exposé précédent

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XVI: 26, 298-313, LNM 920 (1982)

**DELLACHERIE, Claude**; **LENGLART, Érik**

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)

This paper is a sequel to 1524. Let $\Theta$ be a*chronology,* i.e., a family of stopping times containing $0$ and $\infty$ and closed under the operations $\land,\lor$---examples are the family of all stopping times, and that of all deterministic stopping times. The general problem discussed is that of defining an optional process $X$ on $[0,\infty]$ such that for each $T\in\Theta$ $X_T$ is a.s. equal to a given r.v. (${\cal F}_T$-measurable). While in 1525 the discussion concerned supermartingales, it is extended here to processes which satisfy a semi-continuity condition from the right

Keywords: Stopping times

Nature: Original

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XIX: 16, 222-229, LNM 1123 (1985)

**DELLACHERIE, Claude**

Quelques résultats sur les maisons de jeu analytiques

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XXIV: 05, 52-104, LNM 1426 (1990)

**DELLACHERIE, Claude**

Théorie des processus de production

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XXV: 01, 1-9, LNM 1485 (1991)

**DELLACHERIE, Claude**

Théorie non-linéaire du potentiel : Un principe unifié de domination et du maximum et quelques applications

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XXXII: 01, 1-5, LNM 1686 (1998)

**DELLACHERIE, Claude**; **IWANIK, Anzelm**

Sous-mesures symétriques sur un ensemble fini

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XLIII: 06, 187-189, LNM 2006 (2011)

**DELLACHERIE, Claude**

On isomorphic probability spaces

Nature: Original

XLV: 04, 141-157, LNM 2078 (2013)

**DELLACHERIE, Claude**; **ÉMERY, Michel**

Filtrations Indexed by Ordinals; Application to a Conjecture of S. Laurent

Keywords: Filtration

Nature: Original

Un complément au théorème de Weierstrass-Stone (Functional analysis)

An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore

Keywords: Stone-Weierstrass theorem

Nature: Original

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III: 04, 93-96, LNM 88 (1969)

Une application aux fonctionnelles additives d'un théorème de Mokobodzki (Markov processes)

Mokobodzki showed the existence of ``rapid ultrafilters'' on the integers, with the property that applied to a sequence that converges in probability they converge a.s. (see for instance Dellacherie-Meyer,

Comment: See also 203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

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III: 05, 97-114, LNM 88 (1969)

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,

Keywords: Sets with countable sections

Nature: Original

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III: 06, 115-136, LNM 88 (1969)

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 (

Comment: See Dellacherie,

Keywords: Sierpinski's ``rabotages'', Semi-polar sets

Nature: Original

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IV: 05, 60-70, LNM 124 (1970)

Un exemple de la théorie générale des processus (General theory of processes)

In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly

Comment: This example has become classical. See for example Dellacherie-Meyer,

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

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IV: 06, 71-72, LNM 124 (1970)

Au sujet des sauts d'un processus de Hunt (Markov processes)

Two a.s. results on jumps: the process cannot jump

Comment: Both results are improvements of previous results of Meyer and Weil

Keywords: Hunt processes, Semi-polar sets

Nature: Original

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IV: 07, 73-75, LNM 124 (1970)

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

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IV: 08, 76-76, LNM 124 (1970)

Un lemme de théorie de la mesure (Measure theory)

A lemma used by Erdös, Kesterman and Rogers (

Keywords: Convergence in norm, Subsequences

Nature: Original proofs

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IV: 19, 240-282, LNM 124 (1970)

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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V: 07, 77-81, LNM 191 (1971)

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,

Keywords: Capacities

Nature: Original

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V: 08, 82-85, LNM 191 (1971)

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

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V: 09, 86-86, LNM 191 (1971)

Correction à ``Ensembles Aléatoires II'' (Descriptive set theory)

Correction to Dellacherie 306

Comment: See Dellacherie 511

Keywords: Sierpinski's ``rabotages''

Nature: Original

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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

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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

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V: 12, 127-137, LNM 191 (1971)

Un contre-exemple au problème des laplaciens approchés (Martingale theory)

The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years

Comment: Problem originated in Meyer,

Keywords: Submartingales, Supermartingales

Nature: Original

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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

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VII: 04, 36-37, LNM 321 (1973)

Temps d'arrêt totalement inaccessibles (General theory of processes)

Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left

Keywords: Stopping times, Accessible sets, Totally inaccessible stopping times

Nature: Original

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VII: 05, 38-47, LNM 321 (1973)

Sur les théorèmes fondamentaux de la théorie générale des processus (General theory of processes)

This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved

Comment: The idea of this paper has proved fruitful. See for instance Lenglart, 1449; Le Jan,

Keywords: Stopping times, Section theorems

Nature: Original

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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

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VII: 07, 51-57, LNM 321 (1973)

Une conjecture sur les ensembles semi-polaires (Markov processes)

For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets

Comment: The conjecture can be proved, using a general result of Mokobodzki, see 1238

Keywords: Polar sets, Semi-polar sets

Nature: Original

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VII: 08, 58-60, LNM 321 (1973)

Potentiels de fonctionnelles additives. Un contre-exemple de Knight (Markov processes)

An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point

Keywords: Additive functionals

Nature: Exposition, Original additions

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VIII: 04, 22-24, LNM 381 (1974)

Un ensemble progressivement mesurable... (General theory of processes)

The set of starting times of Brownian excursions from $0$ is a well-known example of a progressive set which does not contain any graph of stopping time. Here it is shown that considering the same set for the excursions from any $a$ and taking the union of all $a$, the corresponding set has the same property and has uncountable sections

Comment: Other such examples are known, such as the set of times at which the law of the iterated logarithm fails

Keywords: Progressive sets, Section theorems

Nature: Original

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VIII: 05, 25-26, LNM 381 (1974)

Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)

This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems

Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,

Keywords: Brownian motion, Poisson processes, Previsible representation

Nature: Original

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IX: 08, 239-245, LNM 465 (1975)

Un nouveau théorème de projection et de section (General theory of processes)

Optional section and projection theorems are proved without assuming the ``usual conditions'' on the filtration

Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie 705. The result takes its definitive form in Dellacherie-Meyer,

Keywords: Section theorems, Optional processes, Projection theorems

Nature: Original

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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

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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

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IX: 17, 390-405, LNM 465 (1975)

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,

Keywords: Determinacy of games, Gale and Stewart theorem

Nature: Original

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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

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IX: 28, 494-494, LNM 465 (1975)

Correction à ``Intégrales stochastiques par rapport...'' (General theory of processes)

This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie 805

Comment: Another way of filling this gap is given by Ruiz de Chavez 1821. The same gap for the Poisson process is corrected in 2002

Keywords: Previsible representation

Nature: Original

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IX: 29, 495-495, LNM 465 (1975)

Une propriété des ensembles semi-polaires (Markov processes)

It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)

Keywords: Semi-polar sets

Nature: Original

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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

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X: 30, 545-577, LNM 511 (1976)

Sur la construction de noyaux boréliens (Measure theory)

This answers questions of Getoor 923 and Meyer 924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest

Comment: For a presentation in book form, see Dellacherie-Meyer,

Keywords: Pseudo-kernels, Regularization

Nature: Original

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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

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XI: 04, 34-46, LNM 581 (1977)

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,

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Exposition, Original additions

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XI: 05, 47-50, LNM 581 (1977)

Deux remarques sur la séparabilité optionnelle (General theory of processes)

Optional separability was defined by Doob,

Keywords: Optional processes, Separability, Changes of time

Nature: Original

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XI: 22, 362-364, LNM 581 (1977)

Sur la régularisation des surmartingales (Martingale theory)

It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)

Comment: See 1524

Keywords: General filtrations, Strong supermartingales

Nature: Original

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XI: 23, 365-375, LNM 581 (1977)

Changements de temps et intégrales stochastiques (Martingale theory)

A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135

Comment: The last section states an interesting open problem

Keywords: Changes of time, Spectral representation

Nature: Original

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XII: 10, 70-77, LNM 649 (1978)

A propos du travail de Yor sur le grossissement des tribus (General theory of processes)

This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 12, 98-113, LNM 649 (1978)

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

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XII: 29, 424-424, LNM 649 (1978)

Convergence en probabilité et topologie de Baxter-Chacón (General theory of processes)

It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability

Keywords: Stopping times

Nature: Original

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XII: 30, 425-427, LNM 649 (1978)

Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)

Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$

Keywords: Stopping times

Nature: Original

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XII: 38, 509-511, LNM 649 (1978)

Appendice à l'exposé de Mokobodzki (Measure theory, General theory of processes)

Some comments on 1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from 707

Keywords: Sets with countable sections, Semi-polar sets

Nature: Original

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XII: 39, 512-514, LNM 649 (1978)

Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (General theory of processes)

The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes

Keywords: Essential suprema, Evanescent sets

Nature: Original

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XII: 40, 515-522, LNM 649 (1978)

Supports optionnels et prévisibles d'une P-mesure et applications (General theory of processes)

A $P$-measure is a measure on $\Omega\times

Comment: See 1339 for a complement concerning honest times

Keywords: Projection theorems, Support, Honest times

Nature: Original

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XII: 41, 523-523, LNM 649 (1978)

Erratum et addendum à ``les dérivations en théorie descriptive des ensembles et le théorème de la borne'' (Descriptive set theory)

A few corrections to 1104

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Correction

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XII: 43, 564-566, LNM 649 (1978)

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 suprema

Nature: Original

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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

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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

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XII: 54, 742-745, LNM 649 (1978)

Quelques applications du lemme de Borel-Cantelli à la théorie des semimartingales (Martingale theory, Stochastic calculus)

The general idea is the following: many constructions relative to one single semimartingale---like finding a sequence of stopping times increasing to infinity which reduce a local martingale, finding a change of law which sends a given semimartingale into $H^1$ or $H^2$ (locally)---can be strengthened to handle at the same time countably many given semimartingales

Nature: Original

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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

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XIII: 31, 371-377, LNM 721 (1979)

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

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XV: 24, 320-346, LNM 850 (1981)

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (General theory of processes)

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

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XV: 26, 351-370, LNM 850 (1981)

Mesurabilité des débuts et théorème de section~: le lot à la portée de toutes les bourses (General theory of processes)

One of the main topics in these seminars has been the application to stochastic processes of results from descriptive set theory and capacity theory, at different levels. Since these results are considered difficult, many attempts have been made to shorten and simplify the exposition. A noteworthy one was 511, in which Dellacherie introduced ``rabotages'' (306) to develop the theory without analytic sets; see also 1246, 1255. The main feature of this paper is a new interpretation of rabotages as a two-persons game, ascribed to Telgarsky though no reference is given, leading to a pleasant exposition of the whole theory and its main applications

Keywords: Section theorems, Capacities, Sierpinski's ``rabotages''

Nature: Original

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XV: 27, 371-387, LNM 850 (1981)

Sur les noyaux $\sigma$-finis (Measure theory)

This paper is an improvement of 1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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XVI: 02, 8-28, LNM 920 (1982)

Intégrales de capacités fortement sous-additives

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XVI: 03, 29-40, LNM 920 (1982)

Appendice à l'exposé précédent

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XVI: 26, 298-313, LNM 920 (1982)

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)

This paper is a sequel to 1524. Let $\Theta$ be a

Keywords: Stopping times

Nature: Original

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XIX: 16, 222-229, LNM 1123 (1985)

Quelques résultats sur les maisons de jeu analytiques

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XXIV: 05, 52-104, LNM 1426 (1990)

Théorie des processus de production

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XXV: 01, 1-9, LNM 1485 (1991)

Théorie non-linéaire du potentiel : Un principe unifié de domination et du maximum et quelques applications

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XXXII: 01, 1-5, LNM 1686 (1998)

Sous-mesures symétriques sur un ensemble fini

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XLIII: 06, 187-189, LNM 2006 (2011)

On isomorphic probability spaces

Nature: Original

XLV: 04, 141-157, LNM 2078 (2013)

Filtrations Indexed by Ordinals; Application to a Conjecture of S. Laurent

Keywords: Filtration

Nature: Original