A probabilistic ergodic decomposition result
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, p. 932-942

Let $\left(X,𝔛,\mu \right)$ be a standard probability space. We say that a sub-σ-algebra $𝔅$ of $𝔛$ decomposes μ in an ergodic way if any regular conditional probability ${}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P$ with respect to $𝔅$ and μ satisfies, for μ-almost every xX, $\forall B\in 𝔅,{}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\left(x,B\right)\in \left\{0,1\right\}$. In this case the equality $\mu \left(·\right)={\int }_{X}{}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\left(x,·\right)\mu \left(\mathrm{d}x\right)$, gives us an integral decomposition in “$𝔅$-ergodic” components. For any sub-σ-algebra $𝔅$ of $𝔛$, we denote by $\overline{𝔅}$ the smallest sub-σ-algebra of $𝔛$ containing $𝔅$ and the collection of all sets A in $𝔛$ satisfying μ(A)=0. We say that $𝔅$ is μ-complete if $𝔅=\overline{𝔅}$. Let $\left\{{𝔅}_{i}i\in I\right\}$ 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, ${\bigcap }_{i\in J}\overline{{𝔅}_{i}}=\overline{{\bigcap }_{i\in J}{𝔅}_{i}}$; this assumption is satisfied in particular when the σ-algebras ${𝔅}_{i}$, iI, are μ-complete. Then we prove that the sub-σ-algebra ${\bigcap }_{i\in I}{𝔅}_{i}$ decomposes μ in an ergodic way.

Soit $\left(X,𝔛,\mu \right)$ un espace probabilisé standard. Nous disons qu'une sous-tribu $𝔅$ de $𝔛$ décompose ergodiquement μ si toute probabilité conditionnelle régulière ${}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P$ relativement à $𝔅$ et μ, vérifie, pour μ-presque tout xX, $\forall B\in 𝔅,{}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\left(x,B\right)\in \left\{0,1\right\}$. Dans ce cas l'égalité $\mu \left(·\right)={\int }_{X}{}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\left(x,·\right)\mu \left(\mathrm{d}x\right)$, nous donne une décomposition intégrale en composantes «$𝔅$-ergodiques.» Pour toute sous-tribu $𝔅$ de $𝔛$, nous notons $\overline{𝔅}$ la plus petite sous-tribu de $𝔛$ contenant $𝔅$ et tous les sous-ensembles mesurables de X de μ-mesure nulle. Nous disons que la tribu $𝔅$ est μ-complète si $𝔅=\overline{𝔅}$. Soit $\left\{{𝔅}_{i}i\in I\right\}$ une famille non vide de sous-tribus de $𝔛$ décomposant ergodiquement μ. Supposons que, pour toute partie finie J de I, ${\bigcap }_{i\in J}\overline{{𝔅}_{i}}=\overline{{\bigcap }_{i\in J}{𝔅}_{i}}$; cette hypothèse est satisfaite si les tribus ${𝔅}_{i}$, iI, sont μ-complètes. Alors la sous-tribu ${\bigcap }_{i\in I}{𝔅}_{i}$ décompose ergodiquement μ.

DOI : https://doi.org/10.1214/08-AIHP302
Classification:  28A50,  28D05,  60A10
Keywords: regular conditional probability, disintegration of probability, quasi-invariant measures, ergodic decomposition
@article{AIHPB_2009__45_4_932_0,
author = {Raugi, Albert},
title = {A probabilistic ergodic decomposition result},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {45},
number = {4},
year = {2009},
pages = {932-942},
doi = {10.1214/08-AIHP302},
zbl = {1204.28008},
mrnumber = {2572158},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2009__45_4_932_0}
}

Raugi, Albert. A probabilistic ergodic decomposition result. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, pp. 932-942. doi : 10.1214/08-AIHP302. http://www.numdam.org/item/AIHPB_2009__45_4_932_0/

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