I have the following problem. A random variable $Y$ depends on $X_T$, where $X$ is a strong Markov process generating the filtration of the space, but not on what has happend earlier than $T$. ($Y$ is not independent of $X_T$, but independent of all $X_s 1_{s<T}$, where $s$ is in the real numbers.) Is there a way to show
$E\left[Y|F^X_T\right]=E[Y|X_T]$?
Thank you in advance!