Let $(\Omega,\mathcal{F},P)$ be a probability space and $X$ a random variable. If $\mathcal{G}$ is a $\sigma$-algebra and X is independent of $\mathcal{G}$ we know that $\sigma(X)$ is independent of $\mathcal{G}$.
In this case, is it true that $X$ is independent of every $Y$ which is $\mathcal{G}$-measurable?
I assume yes, since $\sigma(Y)\subseteq{\mathcal{G}}$, am I right?
2026-05-15 05:07:02.1778821622
Random variable independent of $\sigma$-algebra
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