Independence of random variables - measurability

Question

Let's suppose that $X_t$ and $Y_t$ are random variables, which are $\mathcal{F}_t$-measurable, $\mathcal{F}_t$ is a filtration. Suppose also that random variables $Z$ and $Y_t$ are independent. Does it mean that also $Z$ and $X_t$ are independent? If yes, how should I prove it? I'm considering standard probability space. Thank you in advance for your help

Answer 1

No, basically because the mere fact that $X$ and $Y$ are both measurable with respect to a common sigma algebra tells us very little about $X$ and $Y$. For example, if the underlying sigma algebra is the power set, then all variables are measurable, and of course we can find three variables $X$, $Y$ and $Z$ such that $Z$ is independent to $Y$ but not $X$.

Your proposition is true if the underlying sigma algebra is, for example, the sigma algebra generated by $Y$, so that $X$ is $Y$-measurable and therefore more or less (but not really) a function of $Y$. Obviously if $Z$ is independent to $Y$ it's going to be independent to a function of $Y$.