What does the notation $E_{X_n}$ with $X_n$ a random variable refers to?

Question

Context : Conditionnal probabilities : Markov Properties (strong or weak) What is the meaning of the notation $E_{X_n}[Y] $ with $ X_n,Y$ random variables ?

Found many courses using this notation without introduction , does it refer to $E[Y,\sigma(X_n)]$ ?

Answer 1

$E_{X_n}(Y):\Omega \rightarrow \mathbb{R},w\rightarrow E_{X_n(w)}[Y]=E_{P(.|X_0=X_n(w))}[Y]$