Conditional independence of two variables assuming their conditionals are independent

Question

Let $X, Y, Z$ be random variables. It is well-known that $X \mathbin{\bot} Y$ does not imply $X \mathbin{\bot} Y \mathbin{|} Z$ and the converse also does not hold. However, if the random variable $X \mathbin{|}Z$ is independent from $Y\mathbin{|}Z$, would this imply $X \mathbin{\bot} Y\mathbin{|}Z$?

Answer 1

$X|Z$ and $Y|Z$ are not random variables -- under some additional assumptions they might be families of random variables, one for each value of $Z$. I don't think there's anything for "$X|Z$ is independent of $Y|Z$" to mean apart from $X\perp Y|Z$.

That is, $X\perp Y\mid Z$ means "for any suitably-measurable $A$, $B$ and $C$, $$P(X\in A|Z\in C)P(Y\in B\mid Z\in C)=P(X\in A\cap Y\in B\mid Z\in C)"$$

If "$X|Z$ is independent of $Y|Z$" means anything, it also means that.