I am reading Durrett's Probability: Theory and Examples, and trying to understand its context about Markov Chains. It is not hard to understand the theorem and proofs but when it comes to concrete examples, I have a hard time calculate the answer....
Theorem 6.3.1 states that: Let $Y:\Omega_0 \rightarrow\mathbf{R}$ be bounded and measurable. $$ E_\mu(Y\circ\theta_n\mid\mathcal{F}_n)=E_{X_n}Y$$
Now, I am trying to give an example and try to figure out what the theorem really means.
Suppose that we are given a two states markov chain with transition matrix: $$ \begin{pmatrix} 0 & 1 \\ 1/2 & 1/2\\ \end {pmatrix} $$
And the initial distribution $(1,0)$. How can I calculate $E_{X_2}Y$?
Also, how to verify the theorem if $Y=\omega_1+2\omega_2$ and $n=2$?
The term $E_{X_2}Y$, as what I have understood, is the expectation of Y given the initial distribution $X_2$. However, I have no idea the exact calculation is.
Thank you!