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.
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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 ${\mathfrak{B}}=\overline{\mathfrak{B}}$ .
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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.