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