Are the following two the same:
$E[V(X_{t_{k+1}})|g(X_{t_{k+1}}),X_{t_k}]$
and
$E[E[V(X_{t_{k+1}})|g(X_{t_{k+1}})]|X_{t_k}]$
Where $X$ is Markov chain
$X_{t_k} \in \mathcal{R}^n$
$V: \mathcal{R}^n \rightarrow \mathcal{R}$
$g: \mathcal{R}^n \rightarrow \mathcal{R}$
V is some value function.
If not, what is the difference ? Any help would be really appreciated, even in reframing the question
Thanks!