# A probabilistic ergodic decomposition result

• Volume: 45, Issue: 4, page 932-942
• ISSN: 0246-0203

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## Abstract

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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 $𝔅$ andμ satisfies, for μ-almost every x∈X, $\forall B\in 𝔅,{}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\left(x,B\right)\in \left\{0,1\right\}$. 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 setsAin $𝔛$ satisfyingμ(A)=0. We say that $𝔅$ isμ-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 theσ-algebras ${𝔅}_{i}$,i∈I, are μ-complete. Then we prove that the sub-σ-algebra ${\bigcap }_{i\in I}{𝔅}_{i}$ decomposesμ in an ergodic way.

## How to cite

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Raugi, Albert. "A probabilistic ergodic decomposition result." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 932-942. <http://eudml.org/doc/78062>.

@article{Raugi2009,
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 $\{\mathfrak \{B\}\}$ andμ satisfies, for μ-almost every x∈X, $\forall B\in \{\mathfrak \{B\}\},\{\}^\{\mathfrak \{B\}\}\!\!P(x,B)\in \lbrace 0,1\rbrace$. 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 setsAin $\{\mathfrak \{X\}\}$ satisfyingμ(A)=0. We say that $\{\mathfrak \{B\}\}$ isμ-complete if $\{\mathfrak \{B\}\}=\overline\{\mathfrak \{B\}\}$. Let $\lbrace \{\mathfrak \{B\}\}_\{i\}i\in I\rbrace$ 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 theσ-algebras $\{\mathfrak \{B\}\}_\{i\}$,i∈I, are μ-complete. Then we prove that the sub-σ-algebra $\bigcap _\{i\in I\}\{\mathfrak \{B\}\}_\{i\}$ decomposesμ in an ergodic way.},
author = {Raugi, Albert},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {regular conditional probability; disintegration of probability; quasi-invariant measures; ergodic decomposition},
language = {eng},
number = {4},
pages = {932-942},
publisher = {Gauthier-Villars},
title = {A probabilistic ergodic decomposition result},
url = {http://eudml.org/doc/78062},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Raugi, Albert
TI - A probabilistic ergodic decomposition result
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 932
EP - 942
AB - 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 ${\mathfrak {B}}$ andμ satisfies, for μ-almost every x∈X, $\forall B\in {\mathfrak {B}},{}^{\mathfrak {B}}\!\!P(x,B)\in \lbrace 0,1\rbrace$. 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 setsAin ${\mathfrak {X}}$ satisfyingμ(A)=0. We say that ${\mathfrak {B}}$ isμ-complete if ${\mathfrak {B}}=\overline{\mathfrak {B}}$. Let $\lbrace {\mathfrak {B}}_{i}i\in I\rbrace$ 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 theσ-algebras ${\mathfrak {B}}_{i}$,i∈I, are μ-complete. Then we prove that the sub-σ-algebra $\bigcap _{i\in I}{\mathfrak {B}}_{i}$ decomposesμ in an ergodic way.
LA - eng
KW - regular conditional probability; disintegration of probability; quasi-invariant measures; ergodic decomposition
UR - http://eudml.org/doc/78062
ER -

## References

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