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X: 01, 1-18, LNM 511 (1976)
BRÉMAUD, Pierre
La méthode des semi-martingales en filtrage quand l'observation est un processus ponctuel marqué (Martingale theory, Point processes)
This paper discusses martingale methods (as developed by Jacod, Z. für W-theorie, 31, 1975) in the filtering theory of point processes
Comment: The author has greatly developed this topic in his book Poisson Processes and Queues, Springer 1981
Keywords: Point processes, Previsible representation, Filtering theory
Nature: Original
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X: 02, 19-23, LNM 511 (1976)
CHACON, Rafael V.; WALSH, John B.
One-dimensional potential imbedding (Brownian motion)
The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool
Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Skorohod imbedding
Nature: Original
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X: 03, 24-39, LNM 511 (1976)
JACOD, Jean; MÉMIN, Jean
Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)
The natural filtration of a regenerative set $M$ is that of the corresponding age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration
Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second Azéma's martingale'' (not the well known one which has the chaotic representation property)
Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation
Nature: Original
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X: 04, 40-43, LNM 511 (1976)
KAZAMAKI, Norihiko
A simple remark on the conditioned square functions for martingale transforms (Martingale theory)
This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time
Comment: The author has published a paper on a related subject in Tôhoku Math. J., 28, 1976
Keywords: Angle bracket, Inequalities
Nature: Original
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X: 05, 44-77, LNM 511 (1976)
KUNITA, Hiroshi
Absolute continuity for Markov processes (Markov processes)
This paper is devoted to a progressive'' Lebesgue decomposition of the laws of a Markov process with respect to a second one in the same filtration, and the structure of the corresponding density. The two processes are assumed to be Hunt processes, and for part of the paper satisfy Hunt's hypothesis (K) (all excessive functions are regular, or semi-polar sets are polar). The topics discussed are the following: Lévy systems and the relation between the Lévy systems of a process and of its transform by a multiplicative functional; structure of exact perfect terminal times, which are shown to be hitting times of sets in space-time, by the process $(X_{t-},X_t)$ (a version of a result of Walsh-Weil, Ann. Sci. ENS, 5, 1972); the Lebesgue decomposition'' of a Markov process with respect to another, and the fact that if absolute continuity holds on the germ field it also holds up to some maximal terminal time; a condition for this terminal time to be equal to the lifetime, under hypothesis (K)
Comment: The pasting together of the Lebesgue decompositions of a probability measure with respect to another one, on the $\sigma$-fields of a given filtration, is called the Kunita decomposition, and is not restricted to Markov processes. For the general case, see Yoeurp, in LN 1118, Grossissements de filtrations, 1985
Keywords: Absolute continuity of laws, Hunt processes, Terminal times, Kunita decomposition
Nature: Original
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X: 06, 78-85, LNM 511 (1976)
Germ-field Markov property for multiparameter processes (Miscellanea)
The paper studies the relations between several Markov properties of a process indexed by an open set of $R^n$
Comment: To be completed
Keywords: Several parameter Brownian motions, Several parameter processes
Nature: Original
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X: 07, 86-103, LNM 511 (1976)
MEYER, Paul-André
La théorie de la prédiction de F. Knight (General theory of processes)
This paper is devoted to the work of Knight, Ann. Prob. 3, 1975, the main idea of which is to associate with every reasonable process $(X_t)$ another process, taking values in a space of probability measures, and whose value at time $t$ is a conditional distribution of the future of $X$ after $t$ given its past before $t$. It is shown that the prediction process contains essentially the same information as the original process (which can be recovered from it), and that it is a time-homogeneous Markov process
Comment: The results are related to those of Schwartz (presented in 722), the main difference being that the future is predicted instead of the whole path. Knight has devoted to this subject the Essays on the Prediction Process, Hayward Inst. of Math. Stat., 1981, and a book, Foundations of the Prediction Process, Oxford Science Publ. 1992
Keywords: Prediction theory
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X: 08, 104-117, LNM 511 (1976)
MEYER, Paul-André; YOR, Marc
Sur la théorie de la prédiction, et le problème de décomposition des tribus ${\cal F}^{\circ}_{t+}$ (General theory of processes)
This paper contains another version of Knight's theory (preceding paper 1007) for cadlag process instead of measurable processes. These results then are applied to the pathology of germ fields: a natural measurability conjecture does not hold, and an example is given of a process $X_t$ such that its natural $\sigma$-field ${\cal F}_{1+}$ is not generated by ${\cal F}_{1}$ and the germ-field at $0$ of the process $(X_{1+s})$
Comment: On the pathology of germ fields, see H. von Weizsäcker, Ann. Inst. Henri Poincaré, 19, 1983
Keywords: Prediction theory, Germ fields
Nature: Original
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X: 09, 118-124, LNM 511 (1976)
MEYER, Paul-André
Generation of $\sigma$-fields by step processes (General theory of processes)
On a Blackwell measurable space, let ${\cal F}_t$ be a right continuous filtration, such that for any stopping time $T$ the $\sigma$-field ${\cal F}_T$ is countably generated. Then (discarding possibly one single null set), this filtration is the natural filtration of a right-continuous step process
Comment: This answers a question of Knight, Ann. Math. Stat., 43, 1972
Keywords: Point processes
Nature: Original
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X: 10, 125-183, LNM 511 (1976)
MEYER, Paul-André
Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)
This long paper consists of four talks, suggested by E.M.~Stein's book Topics in Harmonic Analysis related to the Littlewood-Paley theory, Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $R^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $R^n\times R_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $R^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (CRAS Paris, 278A, 1974, p.1103) in potential theory and by Kunita (Nagoya M. J., 36, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false
Comment: This paper was rediscovered by Varopoulos (J. Funct. Anal., 38, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253
Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory
Nature: Original
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X: 11, 184-193, LNM 511 (1976)
NAGASAWA, Masao
A probabilistic approach to non-linear Dirichlet problem (Markov processes)
The theory of branching Markov processes in continuous time developed in particular by Ikeda-Nagasawa-Watanabe (J. Math. Kyoto Univ., 8, 1968 and 9, 1969) and Nagasawa (Kodai Math. Sem. Rep. 20, 1968) leads to the probabilistic solution of a non-linear Dirichlet problem
Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 926
Keywords: Branching processes, Dirichlet problem
Nature: Original
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X: 12, 194-208, LNM 511 (1976)
ROST, Hermann
Skorohod stopping times of minimal variance (Markov processes)
Root's (Ann. Math. Stat., 40, 1969) solution of the Shorohod imbedding problem for Brownian motion uses the hitting time of a barrier in space-time. Here Root's construction is extended to general Markov processes, an optimality property of Root's stopping times is proved, as well as the uniqueness of such stopping times
Comment: For previous work of the author on Skorohod's imbedding see Ann. M. Stat. 40, 1969 and Invent. Math. 14, 1971, and in this Seminar 523, 613, 806. A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Skorohod imbedding
Nature: Original
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X: 13, 209-215, LNM 511 (1976)
SEKIGUCHI, Takesi
On the Krickeberg decomposition of continuous martingales (Martingale theory)
The problem investigated is whether the two positive martingales occurring in the Krickeberg decomposition of a $L^1$-bounded continuous martingale of a filtration $({\cal F}_t)$ are themselves continuous. It is shown that the answer is yes only under very stringent conditions: there exists a sub-filtration $({\cal G}_t)$ such that 1) all ${\cal G}$-martingales are continuous 2) the continuous ${\cal F}$-martingales are exactly the ${\cal G}$-martingales
Comment: For related work of the author see Tôhoku Math. J. 28, 1976
Keywords: Continuous martingales, Krickeberg decomposition
Nature: Original
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X: 14, 216-234, LNM 511 (1976)
WILLIAMS, David
The Q-matrix problem (Markov processes)
This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched
Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams Diffusions, Markov Processes and Martingales, vol. 1 (second edition), Wiley 1994. See also 1024
Keywords: Markov chains, Ray compactification, Local times, Excursions
Nature: Original
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X: 15, 235-239, LNM 511 (1976)
WILLIAMS, David
On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)
This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given
Comment: For the original proof of Taylor see Ann. Prob. 3, 1975. For modern references, we should ask Yor
Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula
Nature: Original
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X: 16, 240-244, LNM 511 (1976)
On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (Stochastic calculus, Diffusion theory)
A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions
Comment: This extends to the reflecting barrier case the now classical result in the free'' case due to Yamada-Watanabe, J. Math. Kyoto Univ., 11, 1971. Many of these theorems have now simpler proofs using local times, in the spirit of Revuz-Yor, Continuous Martingales and Brownian Motion, Chapter IX
Keywords: Stochastic differential equations, Boundary reflection
Nature: Original
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X: 17, 245-400, LNM 511 (1976)
MEYER, Paul-André
Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)
This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$
Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books
Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem
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X: 18, 401-413, LNM 511 (1976)
PRATELLI, Maurizio
Sur certains espaces de martingales de carré intégrable (Martingale theory)
The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$
Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250
Keywords: Inequalities, Angle bracket, $BMO$
Nature: Original
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X: 19, 414-421, LNM 511 (1976)
PRATELLI, Maurizio
Espaces fortement stables de martingales de carré intégrable (Martingale theory, Stochastic calculus)
This paper studies closed subspaces of the Hilbert space of square integrable martingales which are stable under optional stochastic integration (see 1018)
Keywords: Stable subpaces, Square integrable martingales, Stochastic integrals, Optional stochastic integrals
Nature: Original
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X: 20, 422-431, LNM 511 (1976)
YAN, Jia-An; YOEURP, Chantha
Représentation des martingales comme intégrales stochastiques des processus optionnels (Martingale theory, Stochastic calculus)
An attempt to build a theory similar to the previsible representation property with respect to a basic local martingale, but using the optional stochastic integral instead of the standard one
Comment: Apparently this optional representation property'' has not been used since
Keywords: Optional stochastic integrals
Nature: Original
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X: 21, 432-480, LNM 511 (1976)
YOEURP, Chantha
Décomposition des martingales locales et formules exponentielles (Martingale theory, Stochastic calculus)
It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $&#279;lta A_t$ do not assume the value $1$. Then this exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality
Comment: The problems in this paper have some relation with Kunita 1005 (in a Markovian set up), and are further studied by Yoeurp in LN 1118, Grossissements de filtrations, 1985. The subject of multiplicative decompositions of positive submartingales is much more difficult since they may vanish. For a simple case see in this volume Yoeurp-Meyer 1023. The general case is due to Azéma (Z. für W-theorie, 45, 1978, presented in 1321) See also 1622
Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures
Nature: Original
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X: 22, 481-500, LNM 511 (1976)
YOR, Marc
Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)
This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up
Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems
Nature: Original
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X: 23, 501-504, LNM 511 (1976)
MEYER, Paul-André; YOEURP, Chantha
Sur la décomposition multiplicative des sousmartingales positives (Martingale theory)
This paper expands part of Yoeurp's paper 1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$
Comment: See the comments on 1021 for the general case. The latter result is related to Meyer 817. For a related paper, see 1203. Further study in 1620
Keywords: Multiplicative decomposition
Nature: Original
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X: 24, 505-520, LNM 511 (1976)
WILLIAMS, David
The Q-matrix problem 2: Kolmogorov backward equations (Markov processes)
This is an addition to 1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required
Keywords: Markov chains
Nature: Original
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X: 25, 521-531, LNM 511 (1976)
BENVENISTE, Albert
Séparabilité optionnelle, d'après Doob (General theory of processes)
A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given
Comment: Doob's original paper appeared in Ann. Inst. Fourier, 25, 1975. See also 1105
Keywords: Optional processes, Separability, Section theorems
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X: 26, 532-535, LNM 511 (1976)
NAGASAWA, Masao
Note on pasting of two Markov processes (Markov processes)
The pasting or piecing out theorem says roughly that two Markov processes taking values in two open sets and agreeing up to the first exit time of their intersection can be extended into a single Markov process taking values in their union. The word roughly'' replaces a precise definition, necessary in particular to handle jumps. Though the result is intuitively obvious, its proof is surprisingly messy. It is due to Courrège-Priouret, Publ. Inst. Stat. Univ. Paris, 14, 1965. Here it is reduced to a revival theorem'' of Ikeda-Nagasawa-Watanabe, J. Math. Kyoto Univ., 8, 1968
Comment: The piecing out theorem is also reduced to a revival theorem in Meyer, Ann. Inst. Fourier, 25,1975
Keywords: Piecing-out theorem
Nature: Original
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X: 27, 536-539, LNM 511 (1976)
KAZAMAKI, Norihiko
A characterization of $BMO$ martingales (Martingale theory)
A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$
Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN 1579, 1994
Keywords: $BMO$
Nature: Original
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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: 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: 31, 578-578, LNM 511 (1976)
MEYER, Paul-André
Un point de priorité (Ergodic theory)
An important remark in Sam Lazaro-Meyer 901, on the relation between Palm measures and the Ambrose-Kakutani theorem that any flow can be interpreted as a flow under a function, was made earlier by F.~Papangelou (1970)
Keywords: Palm measures, Flow under a function, Ambrose-Kakutani theorem
Nature: Acknowledgment
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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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