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I would like to verify the folowing: Let $(X_s)_{s \in [0,\infty)}$ be a stochastic process in continuous time with values in a countable space $E$ and let $\mathcal F_s=\sigma(X_r: 0 \le r \le s)$

Given that for all $n \in \mathbb N $, all $0 \le s_1< ...<s_n<t $ and all $i_1,...,i_n \in E $ such that $P[X_{s_1}=i_1,...X_{s_n } =i_n ]>0$, we have that

$$P[X_t=i|X_{s_1}=i_1,...X_{s_n } =i_n]=P[X_t=i|X_{s_n }=i_n]$$

Then for any $A \in \mathcal B(E)$ and any $s \le t $

$$P[X_t \in A |\mathcal F_s]=P[X_t \in A |\sigma(X_s)]$$

My approach to this was to try to show that given the above $P[X_t \in A |\mathcal F_s]$ is a version of $P[X_t \in A |\sigma(X_s)]$ and that $P[X_t \in A |\sigma(X_s)]$ is a version of $P[X_t \in A |\mathcal F_s]$. Is this correct?

Thus for the first case we have that since $\sigma(X_s) \subset \mathcal F_s $ and from the definition of $P[X_t \in A |\mathcal F_s] $, for any $B \in \sigma(X_s) $

$$E[1_B P[X_t \in A |\mathcal F_s]]=E[1_B 1_{X_t \in A } ]$$ So that one of the two conditions for $P[X_t \in A |\mathcal F_s] $ to be a version of $P[X_t \in A |\sigma(X_s)]$ is satisfied. But I'm not sure of the other condition which is about measurability?

For the second case one immediatly gets the measurability condition but I'm not sure about the "Expectation condition".

Thanks in advance!

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