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,
SubsequencesNature: Original
Retrieve article from Numdam
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 mapsNature: Exposition Retrieve article from Numdam
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,
SubsequencesNature: Original proofs Retrieve article from Numdam
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 logarithmNature: Exposition Retrieve article from Numdam
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 limitNature: Exposition Retrieve article from Numdam
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 spacesNature: Exposition Retrieve article from Numdam
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 setsNature: Original Retrieve article from Numdam
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
41Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
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
1030Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
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,
RegularizationNature: Original Retrieve article from Numdam
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: BimeasuresNature: Original Retrieve article from Numdam
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,
1030Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
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 statisticsNature: Exposition Retrieve article from Numdam
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 theoremNature: Original Retrieve article from Numdam
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,
CapacitiesNature: Original Retrieve article from Numdam
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
1602Keywords: Sets with countable sectionsNature: Original Retrieve article from Numdam
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
707Keywords: Sets with countable sections,
Semi-polar setsNature: Original Retrieve article from Numdam
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 functionsNature: Exposition,
Original additions Retrieve article from Numdam
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 lawNature: New proof of known results Retrieve article from Numdam
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 theoremNature: Original Retrieve article from Numdam
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
1228Keywords: Fuzzy random variables,
Convergence in lawNature: Original Retrieve article from Numdam
