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Consider a probability space $ (\Omega, \mathcal{F}, \mathbb{P}) $ and a countable set of random variables $ \{X_n \mid n \in \mathbb{N}\} $ such that each random variable is independent of a fixed sub sigma-algebra $ \mathcal{G} \subseteq \mathcal{F}. $

Can one conclude that $ \sup_{n \in \mathbb{N}}X_n $ is independent of $ \mathcal{G} $?

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  • $\begingroup$ Did you consider accepting the (full) answer you received below? $\endgroup$ – Did Mar 2 '18 at 16:56
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Consider the space $\Omega=\{0,1\}^2$ consisting of two coin flips with $\mathcal{F}=2^\Omega$ and $\mathbb{P}$ the fair coinflipping measure. Let $\mathcal{G}$ be the $\sigma$-algebra whose informational content is whether these two coin flips agree or not. The canonical projections are independent of $\mathcal{G}$, but their supremum is not: If it is $0$, the coin flips must agree since both must be $0$.

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