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IV: 15, 170-194, LNM 124 (1970)

**MOKOBODZKI, Gabriel**

Densité relative de deux potentiels comparables (Potential theory)

The main problem considered here is the following: given a transient resolvent $(V_{\lambda})$ on a measurable space, a finite potential $Vg$, an excessive function $u$ dominated by $Vg$ in the strong sense (i.e., $Vg-u$ is excessive), show that $u=Vf$ for some $f\leq g$, and compute $f$ by some ``derivation'' procedure, like $\lim_{\lambda\rightarrow\infty} \lambda(I-\lambda V_{\lambda})\,u$

Comment: The main theorem and the technical tools of its proof have been landmarks in the potential theory of a resolvent, though in the case of the resolvent of a good Markov process there is a simple probabilistic proof of the main result. Another exposition can be found in*Séminaire Bourbaki,* **422**, November 1972. See also Chapter XII of Dellacherie-Meyer, *Probabilités et potentiel,* containing new proofs due to Feyel

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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IV: 16, 195-207, LNM 124 (1970)

**MOKOBODZKI, Gabriel**

Quelques propriétés remarquables des opérateurs presque positifs (Potential theory)

A sequel to the preceding paper 415. Almost positive operators are candidates to the role of derivation operators relative to a resolvent

Comment: Same as 415

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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VI: 17, 173-176, LNM 258 (1972)

**MOKOBODZKI, Gabriel**

Pseudo-quotient de deux mesures par rapport à un cône de potentiels. Application à la dualité (Potential theory)

The last four pages of this paper have been omitted by mistake, and appear in the following volume as 729. The general results concerning the axiomatically defined cones of potentials (see for instance the author's exposition in*Séminaire Bourbaki,* 1969-70, **377**) are quickly reviewed first, and then applied to the following problem concerning the potential kernel $V$ of a resolvent: given a pair of measures $\lambda\le\mu$ in the sense of balayage, then we have $\lambda V\le \mu V$ in the ordinary sense. The corresponding density (dominated by $1$) does not depend on the resolvent, but only on the potential cones of excessive functions and potentials associated with it, and a way to compute it is indicated

Keywords: Cones of potentials

Nature: Original

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VII: 29, 319-321, LNM 321 (1973)

**MOKOBODZKI, Gabriel**

Pseudo-quotient de deux mesures, application à la dualité (Potential theory)

Contains the four last pages of 617 omitted from Volume VI

Nature: Correction

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IX: 22, 437-442, LNM 465 (1975)

**MOKOBODZKI, Gabriel**

Relèvement borélien compatible avec une classe d'ensembles négligeables. Application à la désintégration des mesures (Measure theory)

This is a beautiful application of the continuum ``hypothesis'' (axiom). It is shown that if $(E, {\cal E})$ is a separable measurable space (or more generally ${\cal E}$ has the power of the continuum) and ${\cal N}$ is a family of negligible sets within ${\cal E}$, then the space of classes of bounded measurable functions mod.~${\cal N}$ has a linear, isometric, and multiplicative lifting. The proof is rather simple. Essentially the same theorem was discovered independently by Chatterji, see*Vector and Operator Valued Measures,* Academic Press 1973

Comment: The same proof leads to a slightly stronger and useful result (Meyer 2711): if $E$ is a compact metric space and if any two continuous functions equal a.e. are equal everywhere, the lifting can be taken to be the identity on continuous functions, and to be local, i.e., the liftings of two Borel functions equal a.e. in an open set are equal everywhere in this set

Keywords: Continuum axiom, Lifting theorems, Negligible sets

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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XII: 36, 489-490, LNM 649 (1978)

**MOKOBODZKI, Gabriel**

Domination d'une mesure par une capacité (Measure theory)

A bounded measure $\mu$ is said to be dominated by a capacity $C$ (countably subadditive, continuous along increasing sequences; neither strong subadditivity nor decreasing sequences are mentioned) if all sets of capacity $0$ have also measure $0$. The main result then states that the space can be decomposed into a set $A_0$ of capacity $0$, and disjoint sets $A_n$ on each of which $\mu$ is smaller than a multiple of $C$

Keywords: Radon-Nikodym theorem, Capacities

Nature: Original

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XII: 37, 491-508, LNM 649 (1978)

**MOKOBODZKI, Gabriel**

Ensembles à coupes dénombrables et capacités dominées par une mesure (Measure theory, General theory of processes)

Let $X$ be a compact metric space $\mu$ be a bounded measure. Let $F$ be a given Borel set in $X\times**R**_+$. For $A\subset X$ define $C(A)$ as the outer measure of the projection on $X$ of $F\cap(A\times**R**_+)$. Then it is proved that, if there is some measure $\lambda$ such that $\lambda$-null sets are $C$-null (the relation goes the reverse way from the preceding paper 1236!) then $F$ has ($\mu$-a.s.) countable sections, and if the property is strengthened to an $\epsilon-\delta$ ``absolute continuity'' relation, then $F$ has ($\mu$-a.s.) finite sections

Comment: This was a long-standing conjecture of Dellacherie (707), suggested by the theory of semi-polar sets. For further development see 1602

Keywords: Sets with countable sections

Nature: Original

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

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

Intégrales de capacités fortement sous-additives

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XXIII: 26, 324-325, LNM 1372 (1989)

**MOKOBODZKI, Gabriel**

Opérateur carré du champ : un contre-exemple

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XXV: 19, 220-233, LNM 1485 (1991)

**ÉMERY, Michel**; **MOKOBODZKI, Gabriel**

Sur le barycentre d'une probabilité dans une variété (Stochastic differential geometry)

In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$

Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (*J. London Math. Soc.* **46**, 1992), as pointed out in 2650

Keywords: Martingales in manifolds, Convex functions

Nature: Original

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XXVI: 50, 633-633, LNM 1526 (1992)

**ÉMERY, Michel**; **MOKOBODZKI, Gabriel**

Correction au Séminaire~XXV (Stochastic differential geometry)

Points out that the conjecture (due to Émery) at the bottom of page 232 in 2519 is refuted by Kendall (*J. London Math. Soc.* **46**, 1992)

Keywords: Martingales in manifolds

Nature: Correction

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XXVII: 27, 304-311, LNM 1557 (1993)

**MOKOBODZKI, Gabriel**

Représentation d'un semigroupe d'opérateurs sur un espace $L^1$ par des noyaux. Remarques sur deux articles de S.E. Kusnetsov

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XXX: 20, 312-343, LNM 1626 (1996)

**AZÉMA, Jacques**; **JEULIN, Thierry**; **KNIGHT, Frank B.**; **MOKOBODZKI, Gabriel**; **YOR, Marc**

Sur les processus croissants de type injectif

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Densité relative de deux potentiels comparables (Potential theory)

The main problem considered here is the following: given a transient resolvent $(V_{\lambda})$ on a measurable space, a finite potential $Vg$, an excessive function $u$ dominated by $Vg$ in the strong sense (i.e., $Vg-u$ is excessive), show that $u=Vf$ for some $f\leq g$, and compute $f$ by some ``derivation'' procedure, like $\lim_{\lambda\rightarrow\infty} \lambda(I-\lambda V_{\lambda})\,u$

Comment: The main theorem and the technical tools of its proof have been landmarks in the potential theory of a resolvent, though in the case of the resolvent of a good Markov process there is a simple probabilistic proof of the main result. Another exposition can be found in

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

Retrieve article from Numdam

IV: 16, 195-207, LNM 124 (1970)

Quelques propriétés remarquables des opérateurs presque positifs (Potential theory)

A sequel to the preceding paper 415. Almost positive operators are candidates to the role of derivation operators relative to a resolvent

Comment: Same as 415

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

Retrieve article from Numdam

VI: 17, 173-176, LNM 258 (1972)

Pseudo-quotient de deux mesures par rapport à un cône de potentiels. Application à la dualité (Potential theory)

The last four pages of this paper have been omitted by mistake, and appear in the following volume as 729. The general results concerning the axiomatically defined cones of potentials (see for instance the author's exposition in

Keywords: Cones of potentials

Nature: Original

Retrieve article from Numdam

VII: 29, 319-321, LNM 321 (1973)

Pseudo-quotient de deux mesures, application à la dualité (Potential theory)

Contains the four last pages of 617 omitted from Volume VI

Nature: Correction

Retrieve article from Numdam

IX: 22, 437-442, LNM 465 (1975)

Relèvement borélien compatible avec une classe d'ensembles négligeables. Application à la désintégration des mesures (Measure theory)

This is a beautiful application of the continuum ``hypothesis'' (axiom). It is shown that if $(E, {\cal E})$ is a separable measurable space (or more generally ${\cal E}$ has the power of the continuum) and ${\cal N}$ is a family of negligible sets within ${\cal E}$, then the space of classes of bounded measurable functions mod.~${\cal N}$ has a linear, isometric, and multiplicative lifting. The proof is rather simple. Essentially the same theorem was discovered independently by Chatterji, see

Comment: The same proof leads to a slightly stronger and useful result (Meyer 2711): if $E$ is a compact metric space and if any two continuous functions equal a.e. are equal everywhere, the lifting can be taken to be the identity on continuous functions, and to be local, i.e., the liftings of two Borel functions equal a.e. in an open set are equal everywhere in this set

Keywords: Continuum axiom, Lifting theorems, Negligible sets

Nature: Original

Retrieve article from Numdam

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

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

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

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

Keywords: Analytic sets

Nature: Original

Retrieve article from Numdam

XII: 36, 489-490, LNM 649 (1978)

Domination d'une mesure par une capacité (Measure theory)

A bounded measure $\mu$ is said to be dominated by a capacity $C$ (countably subadditive, continuous along increasing sequences; neither strong subadditivity nor decreasing sequences are mentioned) if all sets of capacity $0$ have also measure $0$. The main result then states that the space can be decomposed into a set $A_0$ of capacity $0$, and disjoint sets $A_n$ on each of which $\mu$ is smaller than a multiple of $C$

Keywords: Radon-Nikodym theorem, Capacities

Nature: Original

Retrieve article from Numdam

XII: 37, 491-508, LNM 649 (1978)

Ensembles à coupes dénombrables et capacités dominées par une mesure (Measure theory, General theory of processes)

Let $X$ be a compact metric space $\mu$ be a bounded measure. Let $F$ be a given Borel set in $X\times

Comment: This was a long-standing conjecture of Dellacherie (707), suggested by the theory of semi-polar sets. For further development see 1602

Keywords: Sets with countable sections

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Intégrales de capacités fortement sous-additives

Retrieve article from Numdam

XXIII: 26, 324-325, LNM 1372 (1989)

Opérateur carré du champ : un contre-exemple

Retrieve article from Numdam

XXV: 19, 220-233, LNM 1485 (1991)

Sur le barycentre d'une probabilité dans une variété (Stochastic differential geometry)

In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$

Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (

Keywords: Martingales in manifolds, Convex functions

Nature: Original

Retrieve article from Numdam

XXVI: 50, 633-633, LNM 1526 (1992)

Correction au Séminaire~XXV (Stochastic differential geometry)

Points out that the conjecture (due to Émery) at the bottom of page 232 in 2519 is refuted by Kendall (

Keywords: Martingales in manifolds

Nature: Correction

Retrieve article from Numdam

XXVII: 27, 304-311, LNM 1557 (1993)

Représentation d'un semigroupe d'opérateurs sur un espace $L^1$ par des noyaux. Remarques sur deux articles de S.E. Kusnetsov

Retrieve article from Numdam

XXX: 20, 312-343, LNM 1626 (1996)

Sur les processus croissants de type injectif

Retrieve article from Numdam