Different notations in probability While studying I've come across some different notations that confuses me, so I was hoping you could help me out. 
In the theorem below, I know that $\mathcal{F}_n^X$ is probably the natural filtration generated by $X_0,\ldots,X_n$ - but does this mean $\mathcal{F}_\tau^X$ is the natural filtration generated by $X_0,\ldots, X_\tau$? And what is to be thought of the expected values $\mathbb{E}_\mu$ and $\mathbb{E}_{X_\tau}$? Does this just indicate that we're integrating with respect to another measure than $P$ (from $(\Omega, \mathcal{F}, P)$) - and in that case, how is $X_\tau$ a measure? Or does it have something to do with the initial distribution somehow?

Let $\mu$ be a probability measure on $(S,\mathcal{B}(S))$. Let $Z: S^{\mathbb{N}_0} \to \mathbb{R}$ be
$\mathcal{B}(S)^{\mathbb{N}_0}$-measurable and bounded or non-negative.
(1): For $n \in \mathbb{N}_0; \mathbb{E}_\mu[Z \circ \theta^n \ | \ \mathcal{F}_n^X] = \mathbb{E}_{X_n}[Z]$.
(2): For every $(\mathcal{F}_n^X)_{n\geq 0}$ stopping time $\tau; \mathbb{E}_\mu[Z \circ \theta^n \ | \ \mathcal{F}_\tau^X] = \mathbb{E}_{X_\tau}[Z]$.

 A: For an $(\mathcal{F}^X_n)_n$-stopping time $\tau,$ $\mathcal{F}^X_\tau$ is the stopping time algebra, defined as
$$\mathcal{F}^X_\tau = \{A\in \mathcal{F}: A\cap \{\tau\leq n\}\in \mathcal{F}^X_n\,\,\forall n\in\mathbb{N}_0\}.$$
Intuitively, this corresponds with the information that we have at time $\tau,$ but because $\tau$ is a random variable, this is not the same as the $\sigma$-algebra generated by $X_0, ..., X_\tau,$ because that's a random number of elements, whereas $\sigma$-algebras are deterministic objects.
For the other notation, I suppose this is in the context of Markov chains, because it looks that way. In that case, $\mathbb{E}_x$ denotes expectation with respect to a probability measure $\mathbb{P}^x$ under which $X_0=x$ a.s. That is, expectation if the process starts at $x.$ This is indeed related to the initial distribution: the measure $\mathbb{P}^x$ is the probability measure under which $X_0$ has the initial distribution that is simply $x$ a.s.
As a note: it's good to realize that $\mathbb{E}_x[Z]$ is just some number, for fixed $x$, but that $\mathbb{E}_{X_n}[Z]$ and $\mathbb{E}_{X_\tau}[Z]$ are random variables, because $X_n$ and $X_\tau$ are random variables.
