Markov process in interacting particle systems book 
I have trouble understanding the setup of a Markov process in Liggett's interacting particle systems book.

Let $X$ be a compact metric space with measurable structure given by the $\sigma-$algebra of Borel sets. Let Let $D[0, \infty)$ be the set of all functions $\eta_.$ on $[0, \infty)$ with values in $X$ which are right continuous and have left limits. This is the canonical path space for a Markov process with state space $X$. A Markov process on $X$ is a collection $\{P^\eta, \eta \in X\}$ of probability
measures on $D[0, \infty)$ indexed by $X$. The expectation corresponding to $P^\eta$ will be denoted by $E^\eta$. Thus:
$$E^\eta Z = \int_{D[0,\infty)}ZdP^\eta\tag{1}$$
for any measurable function $Z$ on $D[0,\infty)$ which is integrable relative to $P^{\eta}$. let $C(X)$ be the collection of continuous functions on $X$. For $f\in C(X)$:
$$S(t)f(\eta):=E^\eta f(\eta_t) \tag{2}$$
where $S(t)$ is a linear operator on $C(X)$.
Questions:

$1)$ Let $S$ be a finite set and $X=\{0,1\}^S$ i.e. $X$ is a collection of configurations where each element in $S$ is a $0$ or $1$. In first para, does $\eta_.$ refer to a path through these configurations i.e. a path is an element of $D[0,\infty)$?


$2)$ If the above is right, then is $P^\eta$ a prob measure on paths or on individual configurations? I'm thinking on paths because in $(1)$, $Z$ is a function on $D[0,\infty)$ i.e. $Z:D[0,\infty)\to \mathbb{R}$.


$3)$ If $P^\eta$ is a prob measure on paths, I don't understand what $E^\eta f(\eta_t)$ means in $(2)$ because $\eta_t$ is the configuration at time $t$ and I don't see how $f(\eta_t)$ is a function on $D[0,\infty)$? because as defined in $(1)$, $Z$ has to be a function on $D[0,\infty)$.

 A: Concerning 1): Yes, as stated in the general setting, $\eta_.: \mathbb{R}_+ \ni t \mapsto \eta_t \in X$, so in your situation $\eta_.: t \mapsto \{0,1\}^S$, so indeed, each $\eta_t$ is a configuration.


*Again, as in your general presentation, each $P^{\eta}$ is a measure on $D[0,\infty)$, so it is a probability measure on PATHS of configuration, i.e. on functions $\eta_.$ as above and not on single configurations. The parameter $\eta$ in the notation $P^{\eta}$ intuitively corresponds to the starting point of the measure $P^{\eta}$, i.e. usually $P^{\eta}$ has mass on paths $\gamma: [0,\infty) \to X$ with $\gamma_0 = \eta$.


*The notation in (2) is slightly confusing, which causes your question. Precisely, for each $t \in \mathbb{R}_+$, $S(t): f \mapsto S(t)f$, where $S(t)f: \{0,1\}^S \to \mathbb{R}$ via $S(t)f(\eta) := \int_{\{0,1\}^S}f\circ\pi_tdP^{\eta},$where
$$\pi_t: D[0,\infty) \to \{0,1\}^S, \pi(\eta_.) := \eta_t$$denotes the canonical projection at time $t$. The slightly suggestive notation $f(\eta_t)$ in formula (2) should be read in this spirit: $\eta_.$ denotes a generic element in $D[0,\infty)$ and $\eta_t$ is its value at time $t$. The integral is taken over all paths, projects each path $\eta_.$ to $\eta_t$ and applies $f$ to it. Everything is rigorously well-defined.
I hope this clarifies your issues!
