# Independence of supremum of random variables

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 $\mathcal{G}$?

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 $0$.