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I: 07, 163-165, LNM 39 (1967)

**MEYER, Paul-André**

Sur un théorème de Deny (Potential theory, Measure theory)

In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit

Comment: a version of this result in classical potential theory was proved by Deny,*C.R. Acad. Sci.*, **218**, 1944. The cone of excessive functions possesses good compactness properties, discovered by Mokobodzki. See Dellacherie-Meyer, *Probabilités et Potentiel,* end of chapter XII

Keywords: A.e. convergence, Subsequences

Nature: Original

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III: 01, 1-23, LNM 88 (1969)

**ARTZNER, Philippe**

Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (Measure theory, Functional analysis)

Exposition of three notes by L.~Schwartz (*CRAS* **265**, 1967 and **266**, 1968) showing that some classes of maps between spaces $\ell^p$ and $\ell^q$ transform Gaussian cylindrical measures into Radon measures. The result turns out to be an extension of Minlos' theorem

Comment: Self-contained and detailed exposition, possibly still useful

Keywords: Radonifying maps

Nature: Exposition

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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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VI: 04, 72-89, LNM 258 (1972)

**CHATTERJI, Shrishti Dhav**

Un principe de sous-suites dans la théorie des probabilités (Measure theory)

This paper is devoted to results of the following kind: any sequence of random variables with a given weak property contains a subsequence which satisfies a stronger property. An example is due to Komlós: any sequence bounded in $L^1$ contains a subsequence which converges a.s. in the Cesaro sense. Several results of this kind, mostly due to the author, are presented without detailed proofs

Comment: See 1302 for extensions to the case of Banach space valued random variables. See also Aldous,*Zeit. für W-theorie,* **40**, 1977

Keywords: Subsequences, Central limit theorem, Law of the iterated logarithm

Nature: Exposition

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VII: 19, 198-204, LNM 321 (1973)

**MEYER, Paul-André**

Limites médiales d'après Mokobodzki (Measure theory, Functional analysis)

Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see 304). The same problem is now solved for weak convergences, using a special kind of Banach limits

Comment: The paper contains a few annoying misprints, in particular p.199 line 9*s.c;s.* should be deleted and line 17 *atomique * should be *absolument continu.* For a misprint-free version see Dellacherie-Meyer, *Probabiliés et Potentiel,* Volume C, Chapter X, **55**--57

Keywords: Continuum axiom, Weak convergence of r.v.'s, Medial limit

Nature: Exposition

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IX: 21, 425-436, LNM 465 (1975)

**ÉMERY, Michel**

Primitive d'une mesure sur les compacts d'un espace métrique (Measure theory)

It is well known that the set ${\cal K}$ of all compact subsets of a compact metric space has a natural compact metric topology. The ``distribution function'' of a positive measure on ${\cal K}$ associates with every $A\in{\cal K}$ the measure of the subset $\{K\subset A\}$ of ${\cal K}$. It is shown here (following A.~Revuz,*Ann. Inst. Fourier,* **6**, 1955-56) that the distribution functions of measures are characterized by simple algebraic properties and right continuity

Comment: This elegant theorem apparently never had applications

Keywords: Distribution functions on ordered spaces

Nature: Exposition

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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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IX: 23, 443-463, LNM 465 (1975)

**GETOOR, Ronald K.**

On the construction of kernels (Measure theory)

Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process

Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie 1030. For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **41**

Keywords: Pseudo-kernels, Regularization

Nature: Original

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IX: 24, 464-465, LNM 465 (1975)

**MEYER, Paul-André**

Une remarque sur la construction de noyaux (Measure theory)

With the notation of the preceding report 923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity

Comment: See Dellacherie 1030

Keywords: Pseudo-kernels, Regularization

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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XI: 07, 59-64, LNM 581 (1977)

**HOROWITZ, Joseph**

Une remarque sur les bimesures (Measure theory)

A bimeasure is a function $\beta(A,B)$ of two set variables, which is a measure in each variable when the other is kept fixed. It is important to have conditions under which a bimeasure ``is'' a measure, i.e., is of the form $\mu(A\times B)$ for some measure $\mu$ on the product space. This is known to be true for positive bimeasures ( Kingman,*Pacific J. of Math.,* **21**, 1967, see also 315). Here a condition of bounded variation is given, which implies that a bimeasure is a difference of two positive bimeasures, and therefore is a measure

Comment: Signed bimeasures which are not measures occur naturally, see for instance Bakry 1742, and Émery-Stricker on Gaussian semimartingales

Keywords: Bimeasures

Nature: Original

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XI: 15, 298-302, LNM 581 (1977)

**ZANZOTTO, Pio Andrea**

Sur l'existence d'un noyau induisant un opérateur sous markovien donné (Measure theory)

The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$

Comment: See 923, 924, 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

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XI: 38, 539-565, LNM 581 (1977)

**TORTRAT, Albert**

Désintégration d'une probabilité. Statistiques exhaustives (Measure theory)

This is a detailed review (including proofs) of the problem of existence of conditional distributions

Keywords: Disintegration of measures, Conditional distributions, Sufficient statistics

Nature: Exposition

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XII: 35, 482-488, LNM 649 (1978)

**YOR, Marc**; **MEYER, Paul-André**

Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (Measure theory)

Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability

Comment: The subject is discussed further in 1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the*weak * topology of $L[\infty$, or with the topology of $L^0$

Keywords: Kernels, Radon-Nikodym theorem

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: 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: 58, 770-774, LNM 649 (1978)

**MEYER, Paul-André**

Sur le lemme de La Vallée Poussin et un théorème de Bismut (Measure theory, General theory of processes)

Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces

Keywords: Uniform integrability, Class (D) processes, Moderate convex functions

Nature: Exposition, Original additions

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XV: 02, 6-10, LNM 850 (1981)

**FERNIQUE, Xavier**

Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais (Measure theory)

The theorem asserts the existence, given two probability measures $\mu,\nu$ on a complete separable metric space $(S,d)$, of a measure $\pi$ on $S\times S$ with marginals $\mu$ and $\nu$ such that $\int d(x,y)\,\pi(dx,dy)$ realizes a suitable distance between $\mu$ and $\nu$. An elementary proof is given here by reduction to the compact case

Keywords: Convergence in law

Nature: New proof of known results

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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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XV: 36, 529-546, LNM 850 (1981)

**JACOD, Jean**; **MÉMIN, Jean**

Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (Measure theory)

For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $**R**$. Let us define a fuzzy r.v. $X$ on $(\Omega, {\cal F},P)$ as a probability measure on $\Omega\times**R**$ whose projection on $\Omega$ is $P$. In particular, a standard r.v. $X$ defines such a measure as the image of $P$ under the map $\omega\mapsto (\omega,X(\omega))$. The space of fuzzy r.v.'s is provided with a weak topology, associated with the bounded functions $f(\omega,x)$ which are continuous in $x$ for every $\omega$, or equivalently with the functions $I_A(\omega)\,f(x)$ with $f$ bounded continuous. The main topic of this paper is the study of this topology

Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see 1228

Keywords: Fuzzy random variables, Convergence in law

Nature: Original

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Sur un théorème de Deny (Potential theory, Measure theory)

In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit

Comment: a version of this result in classical potential theory was proved by Deny,

Keywords: A.e. convergence, Subsequences

Nature: Original

Retrieve article from Numdam

III: 01, 1-23, LNM 88 (1969)

Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (Measure theory, Functional analysis)

Exposition of three notes by L.~Schwartz (

Comment: Self-contained and detailed exposition, possibly still useful

Keywords: Radonifying maps

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

VI: 04, 72-89, LNM 258 (1972)

Un principe de sous-suites dans la théorie des probabilités (Measure theory)

This paper is devoted to results of the following kind: any sequence of random variables with a given weak property contains a subsequence which satisfies a stronger property. An example is due to Komlós: any sequence bounded in $L^1$ contains a subsequence which converges a.s. in the Cesaro sense. Several results of this kind, mostly due to the author, are presented without detailed proofs

Comment: See 1302 for extensions to the case of Banach space valued random variables. See also Aldous,

Keywords: Subsequences, Central limit theorem, Law of the iterated logarithm

Nature: Exposition

Retrieve article from Numdam

VII: 19, 198-204, LNM 321 (1973)

Limites médiales d'après Mokobodzki (Measure theory, Functional analysis)

Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see 304). The same problem is now solved for weak convergences, using a special kind of Banach limits

Comment: The paper contains a few annoying misprints, in particular p.199 line 9

Keywords: Continuum axiom, Weak convergence of r.v.'s, Medial limit

Nature: Exposition

Retrieve article from Numdam

IX: 21, 425-436, LNM 465 (1975)

Primitive d'une mesure sur les compacts d'un espace métrique (Measure theory)

It is well known that the set ${\cal K}$ of all compact subsets of a compact metric space has a natural compact metric topology. The ``distribution function'' of a positive measure on ${\cal K}$ associates with every $A\in{\cal K}$ the measure of the subset $\{K\subset A\}$ of ${\cal K}$. It is shown here (following A.~Revuz,

Comment: This elegant theorem apparently never had applications

Keywords: Distribution functions on ordered spaces

Nature: Exposition

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

IX: 23, 443-463, LNM 465 (1975)

On the construction of kernels (Measure theory)

Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process

Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie 1030. For a presentation in book form, see Dellacherie-Meyer,

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

IX: 24, 464-465, LNM 465 (1975)

Une remarque sur la construction de noyaux (Measure theory)

With the notation of the preceding report 923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity

Comment: See Dellacherie 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

XI: 07, 59-64, LNM 581 (1977)

Une remarque sur les bimesures (Measure theory)

A bimeasure is a function $\beta(A,B)$ of two set variables, which is a measure in each variable when the other is kept fixed. It is important to have conditions under which a bimeasure ``is'' a measure, i.e., is of the form $\mu(A\times B)$ for some measure $\mu$ on the product space. This is known to be true for positive bimeasures ( Kingman,

Comment: Signed bimeasures which are not measures occur naturally, see for instance Bakry 1742, and Émery-Stricker on Gaussian semimartingales

Keywords: Bimeasures

Nature: Original

Retrieve article from Numdam

XI: 15, 298-302, LNM 581 (1977)

Sur l'existence d'un noyau induisant un opérateur sous markovien donné (Measure theory)

The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$

Comment: See 923, 924, 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

XI: 38, 539-565, LNM 581 (1977)

Désintégration d'une probabilité. Statistiques exhaustives (Measure theory)

This is a detailed review (including proofs) of the problem of existence of conditional distributions

Keywords: Disintegration of measures, Conditional distributions, Sufficient statistics

Nature: Exposition

Retrieve article from Numdam

XII: 35, 482-488, LNM 649 (1978)

Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (Measure theory)

Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability

Comment: The subject is discussed further in 1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the

Keywords: Kernels, Radon-Nikodym theorem

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: 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

Retrieve article from Numdam

XII: 58, 770-774, LNM 649 (1978)

Sur le lemme de La Vallée Poussin et un théorème de Bismut (Measure theory, General theory of processes)

Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces

Keywords: Uniform integrability, Class (D) processes, Moderate convex functions

Nature: Exposition, Original additions

Retrieve article from Numdam

XV: 02, 6-10, LNM 850 (1981)

Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais (Measure theory)

The theorem asserts the existence, given two probability measures $\mu,\nu$ on a complete separable metric space $(S,d)$, of a measure $\pi$ on $S\times S$ with marginals $\mu$ and $\nu$ such that $\int d(x,y)\,\pi(dx,dy)$ realizes a suitable distance between $\mu$ and $\nu$. An elementary proof is given here by reduction to the compact case

Keywords: Convergence in law

Nature: New proof of known results

Retrieve article from Numdam

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

Retrieve article from Numdam

XV: 36, 529-546, LNM 850 (1981)

Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (Measure theory)

For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $

Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see 1228

Keywords: Fuzzy random variables, Convergence in law

Nature: Original

Retrieve article from Numdam