Let $(X_n)_n$ be homogeneous markov chain. I am trying to understand equation from my textbook:
$$P(A|X_0,X_1,...,X_n)=P(A|X_n)$$ if $A \in \sigma(X_n,X_{n+1},...)$
First of all what what both sides mean?
I think that $P(A|X_n)$ is conditional probability which is random variable such that: $P(A|X_n)=P(A|\sigma(X_n))=E(1_{A}|\sigma(X_n))$ where the last expression is conditional expectation. Similarly left hand side $P(A|X_0,X_1,...X_n)=P(A|\sigma(X_0,X_1,...,X_n))=E(1_A|\sigma(X_0,X_1,...,X_n))$.
$\sigma(X_n)$ is generated by sets {$X_n=k$} for $k$ natural numbers.
Then $P(A|X_n)(w)=P(A|X_n=k)$ when $X_n(w)=k$, similarly $P(A|X_0,X_1,...X_n)(w)=P(A|X_0=k_0,X_1=k_1,...,X_n=k_n)$ when $X_0(w)=k_0,X_1(w)=k_1,...,X_n(w)=k_n$.
So in particular this equation which I am trying to understand says that Markov property holds right? (Just take $A: X_{n+1}=k_{n+1}$).
But it also says that $A$ can be an event such as $X_{n+1}=k_{n+1}, X_{n+2}=k_{n+2}$ or $X_{n+1}=k_{n+1}, X_{n+2}=k_{n+2}, X_{n+3}=k_{n+3}$ or any number of them going forward?
It is more general form of Markov property where $A$ not need to have only one equality in it?
Am I even correct?
If yes then are there any other advantages of this equation compared to Markov property?