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Let $X$ be a random variable and $\{Y_j\}, j\in J$ a family of random variables. $J$ should be an index set, perhaps uncountable. My question is, if $X$ is independent to every finite subfamily of $\{Y_j\}$, i.e. for every $ I \subset J$ and $|I|\in \mathbb{N}$ the family $\{Y_j;j\in I\}$ and $X$ are independent. Could we conclude that $X$ is independent to the whole family $\{Y_j; j\in J\}$?

cheers

math

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Yes, as \[ \mathcal C := \left\{\bigcap_{j \in I} Y_j^{-1}[B_j] \biggm| I \subseteq J\text{ finite}, B_j \subseteq \mathbb R\text{ Borel} \right\} \] is a $\cap$-stable generator of $\sigma(Y_j, j \in J)$ and for $A \in \sigma(X)$ and $C \in \mathcal C$ we have independence of $C$ and $A$.

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    @martini Thx a lot!2012-07-23
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Yes, we can. It all relies on the following result:

If $\mathcal{G}_1,\ldots,\mathcal{G}_n$ are systems of events that are closed under intersection such that $\mathcal{G}_1,\ldots,\mathcal{G}_n$ are independent, then $\sigma(\mathcal{G}_1),\ldots,\sigma(\mathcal{G}_n)$ are also independent.

Since $\sigma((Y_j)_{j\in J})=\sigma((Y_i)_{i\in I}\mid I\subseteq J, |I|\in \mathbb{N})$ the above result gives what you want.