I am trying to understand the notation used in the MC and Stochastic Stability book. Specifically, on page 63.
If I have a markov chain $\Phi$ defined on $(\Omega,\mathcal{F},P_{\mu})$
where $\Omega$=$\prod_{i=0}^{\infty}X_i$ where $X$ is a general state space and $P_\mu$ is the probability measure for $\Phi$ with initial distribution $\mu$ defined for $X$. The notation I am having trouble understanding is the following:
$$\mathbb{E}_\mu[\theta^{n}H|\mathcal{F}^{\Phi}_n]=\mathbb{E}_{\Phi_n}[H]$$
where H is a bounded measurable function on $\Omega$ and $\theta^n$ is the shift operator and $\mathcal{F}^{\Phi}_n=\sigma(\Phi_0,\Phi_1,...,\Phi_n)\subset \mathcal{B}(X^{n+1})$.
Firstly, it is my understanding that $$\mathbb{E}_\mu[H|\mathcal{F}^{\Phi}_n]$$ is a random variable that is $\mathcal{F}^{\Phi}_n$ measurable but agrees with $H=h(\Phi_0,\Phi_1,...)$ on all $A\in\mathcal{F}^{\Phi}_n$ with respect to expectation. But is it not required that $\mathcal{F}^{\Phi}_n\subset \mathcal{F}$ in order for the expectation to make sense?
Secondly, $\theta^{n}H=h(\Phi_n,\Phi_{n+1},...)$ then
$$\mathbb{E}_\mu[\theta^{n}H|\mathcal{F}^{\Phi}_n]=\mathbb{E}_\mu[h(\Phi_n,\Phi_{n+1},...)|\mathcal{F}^{\Phi}_n]$$
I am assuming that somehow independence follows from the sigma algebra and then by time homogeneity of the markov chain then can use $\Phi_n$ as the starting value but this is where I get lost as to what this notation means:
$$\mathbb{E}_{\Phi_n}[H]$$
or the equation $$\mathbb{E}_\mu[\theta^{n}H|\mathcal{F}^{\Phi}_n]$$
in general.
$\mathcal F^\Phi_n$ is $\sigma\{\Phi_0,\Phi_1,\ldots,\Phi_n\}$. Each $A\in \mathcal F^\Phi_n$ can be represented as $\{\omega\in X^\infty: (\Phi_0(\omega,\ldots,\Phi_n(\omega))\in B\}$, where $B\in\mathcal B(X^{n+1})$. Thus $A=B\times X\times X\times\cdots$, so perhaps the text is identifying $A$ with $B$ in thinking of $A$ as a subset of $\mathcal B(X^{n+1})$.