Let $(X_n)_{n \in \mathbb{N}_0}$ be a Markov-chain and $(\mathfrak{F}_n)_{n \in \mathbb{N}_0}$ the induced filtration $\mathfrak{F}_n := \sigma(X_0, \dots, X_n)$. Is then $\mathbb{E}[X_n|\mathfrak{F}_{n-1}] = \mathbb{E}[X_n|\sigma(X_{n-1})]$?
What I mean is, when I actually try to figure out how a certain conditioned expectation of a random variable maps $\Omega$ to $\mathbb{R}$, and the random variable and the $\sigma$-algebra in question happen to meet the above requirements, is it then sufficient, if
$\mathbb{E}[X_n, B] = \mathbb{E}[\mathbb{E}[X_n|\mathfrak{F}_{n-1}]X_n, B] \forall B \in \sigma(X_{n-1})$ (instead of ... $\forall B \in \mathfrak{F}_{n-1}$)?