# 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.

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