A probabilistic ergodic decomposition result

Raugi, Albert

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009), p. 932-942 / Harvested from Numdam

Résumé

Soit $(X,{\mathfrak{X}},\mu)$ un espace probabilisé standard. Nous disons qu'une sous-tribu ${\mathfrak{B}}$ de ${\mathfrak{X}}$ décompose ergodiquement μ si toute probabilité conditionnelle régulière ${}^{\mathfrak{B}}\!\!P$ relativement à ${\mathfrak{B}}$ et μ, vérifie, pour μ-presque tout x∈X, $\forall B\in {\mathfrak{B}},{}^{\mathfrak{B}}\!\!P(x,B)\in\{0,1\}$. Dans ce cas l'égalité $\mu(\cdot)=\int_{X}{}^{\mathfrak{B}}\!\!P(x,\cdot)\mu(\mathrm{d}x)$, nous donne une décomposition intégrale en composantes «${\mathfrak{B}}$-ergodiques.» Pour toute sous-tribu ${\mathfrak{B}}$ de ${\mathfrak{X}}$, nous notons $\overline{\mathfrak{B}}$ la plus petite sous-tribu de ${\mathfrak{X}}$ contenant ${\mathfrak{B}}$ et tous les sous-ensembles mesurables de X de μ-mesure nulle. Nous disons que la tribu ${\mathfrak{B}}$ est μ-complète si ${\mathfrak{B}}=\overline{\mathfrak{B}}$. Soit $\{{\mathfrak{B}}_{i}\dvt i\in I\}$ une famille non vide de sous-tribus de ${\mathfrak{X}}$ décomposant ergodiquement μ. Supposons que, pour toute partie finie J de I, $\bigcap_{i\in J}\overline{{\mathfrak{B}}_{i}}=\overline{\bigcap_{i\in J}{\mathfrak{B}}_{i}}$; cette hypothèse est satisfaite si les tribus ${\mathfrak{B}}_{i}$, i∈I, sont μ-complètes. Alors la sous-tribu $\bigcap_{i\in I}{\mathfrak{B}}_{i}$ décompose ergodiquement μ.

Abstract

Let $(X,{\mathfrak{X}},\mu)$ be a standard probability space. We say that a sub-σ-algebra ${\mathfrak{B}}$ of ${\mathfrak{X}}$ decomposes μ in an ergodic way if any regular conditional probability ${}^{\mathfrak{B}}\!\!P$ with respect to ${\mathfrak{B}}$ and μ satisfies, for μ-almost every x∈X, $\forall B\in{\mathfrak{B}},{}^{\mathfrak{B}}\!\!P(x,B)\in\{0,1\}$. In this case the equality $\mu(\cdot)=\int_{X}{}^{\mathfrak{B}}\!\!P(x,\cdot)\mu(\mathrm{d}x)$, gives us an integral decomposition in “${\mathfrak{B}}$-ergodic” components. For any sub-σ-algebra ${\mathfrak{B}}$ of ${\mathfrak{X}}$, we denote by $\overline{\mathfrak{B}}$ the smallest sub-σ-algebra of ${\mathfrak{X}}$ containing ${\mathfrak{B}}$ and the collection of all sets A in ${\mathfrak{X}}$ satisfying μ(A)=0. We say that ${\mathfrak{B}}$ is μ-complete if ${\mathfrak{B}}=\overline{\mathfrak{B}}$. Let $\{{\mathfrak{B}}_{i}\dvt i\in I\}$ be a non-empty family of sub-σ-algebras which decompose μ in an ergodic way. Suppose that, for any finite subset J of I, $\bigcap_{i\in J}\overline{{\mathfrak{B}}_{i}}=\overline{\bigcap_{i\in J}{\mathfrak{B}}_{i}}$; this assumption is satisfied in particular when the σ-algebras ${\mathfrak{B}}_{i}$, i∈I, are μ-complete. Then we prove that the sub-σ-algebra $\bigcap_{i\in I}{\mathfrak{B}}_{i}$ decomposes μ in an ergodic way.

Détail

Publié le : 2009-01-01

MR 2572158 | Zbl 1204.28008

DOI : https://doi.org/10.1214/08-AIHP302
Classification:  28A50,  28D05,  60A10

Bibliographie

[1] D. L. Burkholder and Y. S. Chow. Iterates of conditional expectations operators. Proc. Amer. Math. Soc. 12 (1961) 490-495. | MR 142144 | Zbl 0106.33201

[2] J.-P. Conze and A. Raugi. On the ergodic decomposition for a cocycle. Preprint, 2007. (pdf version “ReducErg.pdf” in personal university site.) | MR 2539552

[3] G. Greschonig and K. Schmidt. Ergodic decomposition of quasi-invariant probability measures. Colloq. Math. 84/85 (2000) 495-514. | MR 1784210 | Zbl 0972.37003

[4] J. Kerstan and A. Wakolbinger. Ergodic decomposition of probability laws. Z. Wahrsch. Verw. Gebiete 56 (1981) no. 3 399-414. | MR 621119 | Zbl 0444.60004

[5] J. Neveu. Bases Mathématiques du Calcul des Probabilités. Masson, Paris, 1964. | MR 198504 | Zbl 0137.11203

[6] K. Schmidt. A probabilistic proof of ergodic decomposition. Sankhyā Ser. A 40 (1978) no. 1 10-18. | MR 545459 | Zbl 0412.60004

[7] H. Shimomura. Ergodic decomposition of quasi-invariant measures. Publ. Res. Inst. Math. Sci. 14 (1978) no. 2 359-381. | MR 509194 | Zbl 0391.60004

[8] H. Shimomura. Remark to the ergodic decomposition of measures. Publ. Res. Inst. Math. Sci. 26 (1990) no. 5 861-865. | MR 1082320 | Zbl 0716.28005