I need to prove:
$$P(X_t|\mathcal{F}_s) = P(X_t|X_s) \Leftrightarrow E[f(X_t)|\mathcal{F}_s] = E[f(X_t)|X_s]$$
where
$$\mathcal{F}_s= \sigma(X_u,u\leq s)$$
(I have to prove that this two definition of marvok process are equal)
I can see that $$P(X_t|\mathcal{F}_s) = P(X_t|X_s) \rightarrow E[X_t|\mathcal{F}_s] = E[X_t|X_s]$$
But I don't know how to continue from here. Can you help me?